Module MA3429: Differential Geometry
Michaelmas Term 2010
Part III: Sections 8 to 11
David R. Wilkins
c David R. Wilkins 2010
Copyright Contents
8 Connections, Curvature, and Torsion
8.1 Connections on Smooth Vector Bundles . . . . . . . . . . . .
8.2 Curvature of Connections on Vector Bundles . . . . . . . . .
8.3 Induced Connections on Dual Bundles . . . . . . . . . . . .
8.4 Induced Connections on Tensor Products of Vector Bundles
8.5 Affine Connections on Smooth Manifolds . . . . . . . . . . .
8.6 Covariant Derivatives of Tensor Fields . . . . . . . . . . . .
8.7 The First Bianchi Identity . . . . . . . . . . . . . . . . . . .
8.8 The Second Bianchi Identity . . . . . . . . . . . . . . . . . .
137
. 137
. 145
. 154
. 155
. 158
. 164
. 171
. 172
9 Riemannian and Pseudo-Riemannian Manifolds
9.1 Riemannian and Pseudo-Riemannian Metrics . . . .
9.2 The Levi-Civita Connection . . . . . . . . . . . . .
9.3 The Riemann Curvature Tensor . . . . . . . . . . .
9.4 The Sectional Curvatures of a Riemannian Manifold
.
.
.
.
174
. 174
. 178
. 183
. 186
.
.
.
.
188
. 188
. 188
. 189
. 192
.
.
.
.
.
.
.
.
10 Covariant Derivatives along Curves and Surfaces
10.1 Vector Fields along Smooth Maps . . . . . . . . . . . .
10.2 Moving Frames . . . . . . . . . . . . . . . . . . . . . .
10.3 Covariant Differentiation of Vector Fields along Curves
10.4 Vector Fields along Parameterized Surfaces . . . . . . .
i
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
11 Geodesics and Jacobi Fields
11.1 Geodesics . . . . . . . . . . . . . . . . . .
11.2 The First Variations of Length and Energy
11.3 Jacobi Fields . . . . . . . . . . . . . . . .
11.4 The Second Variation of Energy . . . . . .
ii
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
196
196
199
202
204
8
Connections, Curvature, and Torsion
8.1
Connections on Smooth Vector Bundles
Definition Let πE : E → M be a smooth vector bundle over a smooth manifold M . A smooth connection on M is a differential operator D which, at
each point p of M , associates to each smooth section s of the vector bundle
defined around the point p and to each tangent vector Xp at p an element element DXp s of the fibre Ep of the vector bundle over p, where this differential
operator has the following properties:—
(i) DWp +Xp s = DWp s + DXp s
for all tangent vectors Wp and Xp at a point p of M , and for all smooth
sections s of the vector bundle defined around p;
(iii) Dc Xp s = c DXp s
for all real numbers c and tangent vectors Xp at a point p of M , and
for all smooth sections s of the vector bundle defined around p;
(iii) DXp (s + t) = DXp s + DXp t
for all tangent vectors Xp at a point p of M , and for all smooth sections s
and t of the vector bundle defined around p;
(iv) DX (f s) = X[f ] s + f DX s
for all tangent vectors Xp at a point p of M , for all smooth real-valued
functions f defined around p, and for all smooth sections s of the vector
bundle defined around p;
(v) given a smooth vector fields X defined over an open subset U of M ,
and given a smooth section s: U → E of the vector bundle πE : E → M
defined over U , the function that sends points p of U to DXp s is itself
a smooth section of the vector bundle defined over U .
The element DXp s of the fibre Ep of the vector bundle at a point p of the
manifold determined by a tangent vector Xp at p and a smooth section s of
the vector bundle defined around p is referred to as the covariant derivative
of the section s along the tangent vector Xp (with respect to the smooth
connection D).
Example Let M be a smooth n-dimensional submanifold of k-dimensional
Euclidean space Rk , and let π: M × Rk → M be the product bundle over M
with fibre Rk . The tangent space Tp M at each point p of M can be identified
with a vector subspace of Rk . Let
T M = {(p, v) ∈ M × R : v ∈ Tp M }.
137
and let πT M : T M → M be the function defined such that πT M (p, v) = p for all
(p, v) ∈ T M . Then T M is a smooth submanifold of M ×Rk . πT M : T M → M
is a smooth map, and moreover πT M : T M → M is the projection map of a
smooth vector bundle over M with total space T M which is a subbundle of
the product bundle π: M ×Rk → M . This smooth vector bundle πT M : T M →
M can be identified with the tangent bundle of the smooth manifold M .
Let Np M denote the orthogonal complement of Tp M in Rk , and let
N M = {(p, v) ∈ M × R : v ∈ Np M }.
and let πN M : N M : M be the function defined such that πN M (p, v) = p for
all (p, v) ∈ N M . Then N M is also a smooth submanifold of M × Rk .
πN M : N M → M is a smooth map, and moreover πN M : N M → M is the
projection map of a smooth vector bundle over M with total space N M
which is also a subbundle of the product bundle π: M × Rk → M . This
smooth vector bundle πN M : N M → M is referred to as the normal bundle
of the smooth submanifold M of Rk . Its fibre at a point p of M is the
vector space consisting of all vectors v in the ambient Euclidean space Rk
that are orthogonal to the smooth submanifold M . A section of this bundle
is a normal vector field defined on the submanifold M of Rk . The product
bundle over M with fibre Rk is then the direct sum T M ⊕N M of the tangent
bundle T M and the normal bundle N M of M .
Let V: M → Rk be a smooth function from M to Rk . Then there are
smooth real-valued functions V 1 , V 2 , . . . , V k on M such that
V(p) = (V 1 (p), V 2 (p), . . . , V k (p))
for all p ∈ M . Given any smooth (tangential) vector field X on M , we define
∂X V = (X[V 1 ], X[V 2 ], . . . , X[V k ])
for all p ∈ M , where X[V j ] denotes the smooth real-valued function on
M whose value at any point p of M is the directional derivative Xp [V j ] of
the smooth real-valued function V j along the tangent vector Xp at p for
j = 1, 2, . . . , k. Thus if y 1 , y 2 , . . . , y n are smooth local coordinates defined
n
X
∂
over some open subset U of M , and if X =
ui i , then
∂y
i=1
j
X[V ] =
n
X
i=1
ui
∂V j
∂y i
over the open set U . It is easy to verify that the differential operator ∂ defined
as above represents a smooth connection on the product bundle π: M ×Rk →
138
M . Indeed
∂X (V + W) = ∂X V + ∂X W,
∂X+Y V = ∂X V + ∂Y V
and
∂f X V = f ∂X V
for all smooth real-valued functions f and smooth vector fields X and Y on
M , and for all smooth functions V and W from M to Rk . Also
∂X (f V) =
=
=
=
(X[f · V 1 ], . . . , X[f · V k ])
(X[f ] · V 1 + f · X[V 1 ], . . . , X[f ] · V k + f · X[V k ])
X[f ](V 1 , . . . , V k ) + f (X[V 1 ], . . . , X[V k ])
X[f ]V + f ∂X V
where
V(p) = (V 1 (p), . . . , V k (p))
for all p ∈ M .
Let V: M → T M and X: M → T M be smooth sections of the tangent
bundle of M representing smooth tangential vector fields on M , and let
Q: M → T N be a smooth section of the normal bundle of M . These sections
V, X and Q are then smooth sections of the product bundle over M with
fibre Rk , and therefore so are ∂X V and ∂X Q. However ∂Xp V does not in
general belong to the tangent space Tp M at the point p, and ∂Xp Q does
not in general belong to the normal bundle at the point p. We can however
decompose the smooth sections of the product bundle π: M × Rk → M
represented by ∂X V and ∂X V into their tangential and normal components,
so that
∂X V = ∇X V − S(X, V),
∂X Q = DX Q + Š(X, Q),
where
(∇X V)p ∈ Tp M,
(DX Q)p ∈ Np M,
S(X, V)p ∈ Np M,
and Š(X, Q)p ∈ Tp M
for all p ∈ M . Then the tangential components ∇X V and Š(X, Q) are
smooth sections of the tangent bundle πT M : T M → M of M and the normal
components DX Q and S(X, V) are smooth sections of the normal bundle
πN M : N M → M of M . Moreover
∇X (V + W) = ∇X V + ∇X W,
∇f X V = f ∇X V,
∇X+Y V = ∇X V + ∇Y V,
∇X (f V) = X[f ]V + f ∇X V,
139
S(X, V + W) = S(X, V) + S(X, W),
S(X + Y, V) = S(X, V) + S(Y, V),
S(f X, V) = S(X, f V) = f S(X, V),
DX (Q + R) = DX Q + DX R,
Df X Q = f DX Q,
DX+Y Q = DX Q + DY Q,
DX (f Q) = X[f ]Q + f DX Q,
and
Š(X, Q + R) = Š(X, Q) + Š(X, R),
Š(X + Y, Q) = Š(X, Q) + Š(Y, Q),
Š(f X, Q) = Š(X, f Q) = f Š(X, Q),
for all smooth real-valued functions f on M , smooth tangential vector fields
V, W, X and Y on M , and smooth normal vector fields Q and R on M . It
follows that the differential operator ∇ is a smooth connection on the tangent
bundle πT M : T M → M of M . Moreover
X[V.W] = (∇X V).W + V.(∇X W)
for all smooth tangential vector fields X, V and W on M . Similarly the differential operator D is a smooth connection on the normal bundle πN M : N M →
M of M . Moreover
X[Q.R] = (DX Q).R + Q.(DX R)
for all smooth tangential vector fields X and normal vector fields Q and R
on M .
If V is a smooth tangential vector field and Q is a smooth normal vector
field then V and Q are orthogonal at each point of M , and therefore V.Q = 0.
On differentiating this equation we find that
0 = ∂X V.Q + V.∂X Q = −S(X, V).Q + V.Š(X, Q),
and thus
V.Š(X, Q) = S(X, V).Q
for all tangential vector fields X, Y and V and normal vector fields Q on M .
An application of Proposition 6.15 shows that the value of S(X, V) at
any point p of M is determined by the values of the tangential vector fields
X and V at the point p, and therefore the operator S is determined by a
smooth section of the smooth vector bundle N M ⊗ T ∗ M ⊗ T ∗ M over M ,
where πT ∗ M : T ∗ M → M is the cotangent bundle of M . Similarly the operator
140
Š is determined by a smooth section of T M ⊗ T ∗ M ⊗ N ∗ M over M , where
πN ∗ M : N ∗ M → M is the dual of the normal bundle of M . The operator S is
referred to as the second fundamental tensor of the submanifold M of Rk .
The smooth connection ∇ on the tangent bundle of the submanifold M of
Rk is the Levi-Civita connection of this submanifold. One can prove that it
is determined by the inner product on the tangent spaces of the submanifold
that is the restriction to these tangent spaces of the scalar product on Rk .
Indeed a straightforward if lengthy computation establishes that
1
X[Y.Z] + Y[X.Z] − Z[X.Y]
(∇X Y).Z =
2
+ [X, Y].Z − [X, Z].Y − [Y, Z].X
for all tangential vector fields X, Y and Z on M . This identity is a special case of a more general identity applicable to Riemannian and pseudoRiemannian manifolds.
Proposition 8.1 Let D be a smooth connection on a smooth vector bundle
πE : E → M of rank r, and let U be an open subset in M which is contained
in the domain of a smooth coordinate system (x1 , x2 , . . . , xn ) for M and over
which are defined smooth sections e1 , e2 , . . . , er of the vector bundle whose
values e1 (p), e2 (p), . . . , er (p) at each point p of Ep constitute a basis of the
fibre Ep of this vector bundle over the point p. Let
Dj eβ = D
∂
∂xj
eβ =
r
X
Aα β j eα ,
α=1
for j = 1, 2, . . . , n and β = 1, 2, . . . , r, where each function Aα β j is a smooth
real-valued function on U . Let X be a smooth vector field on U , and let
s: U → E be a smooth section of the vector bundle πE : E → M defined over
U , and let
r
n
X
X
j ∂
and s =
f α eα ,
X=
v
j
∂x
α=1
j=1
where v 1 , v 2 , . . . , v n and f 1 , f 2 , . . . , f r are smooth real-valued functions on U .
Then
n
X
DX s =
v j Dj s,
j=1
where
Dj s =
r
X
α=1
r
∂f α X α
+
A β jf β
∂xj
β=1
141
!
eα .
Proof Property (v) in the definition of smooth connections ensures that the
functions Aαβ j are smooth. It follows from properties (i) and (ii) in the
definition of smooth connections that
DX s =
n
X
v j Dj s,
where Dj s = D
j=1
∂
∂xj
s.
It then follows from properties (iii) and (iv) in the definition of smooth connections that
!
r
r
X
X
α
Dj s = Dj
f eα =
Dj (f α eα )
α=1
α=1
r
X
r
X
∂f α
eα +
f β Dj eβ
j
∂x
α=1
β=1
!
r
r
X ∂f α X
+
Aα β j f β eα ,
=
j
∂x
α=1
β=1
=
as required.
Corollary 8.2 Let D be a smooth connection on a smooth vector bundle
πE : E → M of rank r, and let U be an open subset in M which is contained
in the domain of a smooth coordinate system (x1 , x2 , . . . , xn ) for M and over
which are defined smooth sections e1 , e2 , . . . , er and ê1 , ê2 , . . . , êr , where the
values of the sections e1 , e2 , . . . , er at each point p of U constitute a basis of
the fibre Ep of the vector bundle over the point p, and where the values of
the sections ê1 , ê2 , . . . , êr at each point p of U also constitute a basis of the
fibre Ep . Let Aj : U → Mr (R) and Âj : U → Mr (R) the smooth functions from
the open set U to the algebra Mr (R) of r × r matrices with real coefficients
whose values at p ∈ U are the matrices whose entry in row α and column β
are Aα β j (p) and Âα β j (p) respectively, where
Dj eβ = D
∂
∂xj
eβ =
r
X
Aα β j eα ,
α=1
for j = 1, 2, . . . , n and β = 1, 2, . . . , r, and
Dj êη = D
∂
∂xj
êη =
r
X
Âξ η k êξ ,
ξ=1
for j = 1, 2, . . . , n and η = 1, 2, . . . , r. Also let S: U → GL(k, R) the smooth
function from U to the group GL(k, R) of non-singular r × r matrices with
142
real coefficients whose value at p ∈ U is the non-singular matrix whose value
in row α and column ξ is S α ξ (p), where
êξ (p) =
r
X
S α ξ (p)eα (p).
α=1
Then
Âj = S −1 Aj S + S −1
∂S
.
∂xj
Proof It follows from Proposition 8.1 that
r
r
X
X
Dj êη =
∂S α η
Aα β j S β η +
∂xj
β=1
α=1
!
eα .
Now
r
X
(S
r
X
−1 ξ
) α êξ (p) =
ξ=1
(S
−1 ξ
)
β
α Sξ eβ (p)
=
ξ,β=1
r
X
δαβ eβ (p) = eα (p),
β=1
where (S −1 )ηα is the function whose value at any point p of U is the entry
in row η and column α of the matrix S −1 (p) that is the inverse of S(p), and
where δαβ denotes the Kronecker delta that is equal to 1 when α = β but is
equal to zero otherwise. It follows that
!
r
r
r
X
X
X
∂S α η
−1 ξ
α
β
Dj êη =
(S ) α
A β jS η +
Âξ η j êξ ,
êξ =
j
∂x
α,ξ=1
β=1
ξ=1
where
Âξ η j =
r
X
(S −1 )ξ α Aα β j S β η +
r
X
(S −1 )ξ α
α=1
α,β=1
Thus
Âj = S −1 Aj S + S −1
∂S α η
.
∂xj
∂S
,
∂xj
as required.
Corollary 8.3 Let D be a smooth connection on a smooth vector bundle
πE : E → M of rank r, and let U be an open subset in M over which are
defined smooth sections e1 , e2 , . . . , er of the vector bundle πE : E → M whose
values at each point p of Ep constitute a basis of the fibre Ep of this vector
143
bundle over the point p. Then there exist smooth 1-forms ω α β on U for
α, β = 1, 2, . . . , r such that
DXp eβ =
r
X
ω α β (Xp )eα (p)
α=1
for β = 1, 2, . . . , r and for all tangent vectors Xp at points p of U . If s: U → E
is a smooth section of πE : E → M defined over U , and if
s=
n
X
f α eα ,
α=1
where f 1 , f 2 , . . . , f r are smooth real-valued functions on U , then
*
+
r
r
X
X
α
α
β
DXp s =
df +
ω β f , Xp eα ,
α=1
β=1
for all tangent vectors Xp at points p of U . Moreover if ê1 , ê2 , . . . , êr are
smooth sections of the vector bundle πE : E → M over U whose values at
each point p of U also constitute a basis of the fibre Ep of the vector bundle
over p, if ω̂ ξ η are smooth 1-forms on U defined such that
DX êν =
r
X
ω̂ ξ η (X)êξ
ξ=1
and if
êξ (p) =
r
X
S α ξ (p)eα (p),
α=1
where the values of the smooth real-valued functions S α ξ at each point p of
U are the components of a non-singular r × r matrix S(p), then
ω̂
ξ
η
=
r
X
(S
−1 ξ
) αω
α
β
βS η
+
r
X
(S −1 )ξ α dS α η .
α=1
α,β=1
where the values of the smooth real-valued functions (S −1 )ξ α at each point p
of U are the components of the inverse whose inverse S −1 (p) of the matrix S(p). Thus if ω and ω̂ denote the r × r matrices of smooth 1-forms
whose components are the 1-forms ω α β and ω̂ ξ η , then
ω̂ = S −1 ωS + S −1 dS,
where dS denotes the differential of the smooth matrix-valued function
S: U → GL(k, R).
144
Proof Let ωβα =
n
P
Aα β j dxj and ω̂ηξ =
n
P
Âξ η j dxj where Aα β j and Âξ η j
j=1
j=1
are defined as in the statement of Corollary 8.2. Then the identities given in
the statement of this corollary are restatements of those of Proposition 8.1
and Corollary 8.2. They can also be verified by direct calculation.
8.2
Curvature of Connections on Vector Bundles
Let πE : E → M be a smooth vector bundle of rank r over a smooth manifold M , and, for each p ∈ M , let Ep denote the fibre of this bundle over
the point p. Let U be an open set in M , let (x1 , x2 , . . . , xn ) be a smooth
coordinate system for M defined over U , and let e1 , e2 , . . . , er be smooth
sections of πE : E → M over U , where, for each point p of U , the elements
e1 (p), e2 (p), . . . , er (p) constitute a basis of the real vector space Ep .
Let D be a smooth connection on this vector bundle πE : E → M , and let
∂j f =
∂f
∂xj
and Dj s = D
∂
∂xj
s
for all smooth real-valued functions f and for all smooth sections s of the
vector bundle defined over U Then there are smooth functions Aα β j defined
over U such that
r
X
Dj eβ =
Aα β j eα ,
α=1
for β = 1, 2, . . . , r. Let X be a smooth vector field on U , and let s: U → E
be a smooth section of the vector bundle πE : E → M defined over U , and let
X=
n
X
vj
j=1
∂
,
∂xj
Y =
n
X
wj
j=1
∂
∂xj
and s =
n
X
f α eα ,
α=1
where v 1 , v 2 , . . . , v n and f 1 , f 2 , . . . , f r are smooth real-valued functions on U .
Then
n
n
X
X
j
DX s =
v Dj s and DY s =
wj Dj s,
j=1
where
Dj s =
j=1
r
X
∂j f α +
α=1
β=1
(see Proposition 8.1). Then
DX (DY s) =
n
X
r
X
v j Dj (wk Dk s)
j,k=1
145
!
Aα β j f β
eα
=
n X
r
X
v j Dj
wk
∂k f α +
j,k=1 α=1
=
n X
r
X
v j (∂j wk ) ∂k f α +
r
X
=
n
X
Aα γ k f γ
!
eα
!
Aα γ k f γ
eα
γ=1
n
X
r
X
v j wk ∂j
∂k f α +
j,k=1 α=1
+
!
γ=1
j,k=1 α=1
+
r
X
n X
r
X
r
X
!
Aα γ k f γ
eα
γ=1
v j wk
∂k f β +
r
X
!
Aβ γ k f γ
Dj eβ
γ=1
j,k=1 β=1
r
X
j k
v w (∂j ∂k f α )eα
j,k=1 α=1
+
+
+
r
n X
X
j,k=1 α=1
r
n X
X
j,k=1 α=1
r
n X
X
v j (∂j wk ) ∂k f α +
=
n
X
!
Aα γ k f γ
eα
γ=1
j
v w
k
v j wk
j,k=1 α=1
+
r
X
r
X
γ=1
r
X
(∂j Aα γ k )f γ eα
Aα γ k (∂j f γ )eα
γ=1
r
n
X
X
v j w k Aα β j
∂k f β +
r
X
!
Aβ γ k f γ
eα
γ=1
j,k=1 α,β=1
r
X
(∂j ∂k f α )v j wk eα
j,k=1 α=1
+
+
n X
r
X
v j (∂j wk ) ∂k f α +
j,k=1 α=1
n
r
X
X
r
X
!
Aα γ k f γ
eα
γ=1
(Aα γ k ∂j f γ + Aα γ j ∂k f γ ) v j wk eα
j,k=1 α,γ=1
n X
r
r
r
X
X
X
α
+
(∂j A γ k ) +
Aα β j Aβ γ k
j,k=1 α=1
γ=1
!
v j wk f γ eα
β,γ=1
We see that the value of DX (DY s) at a point p of U is determined by the
values of the components v j , wk and f α of X, Y and s at the point p, the
first order partial derivatives of all these components, and the second order
146
partial derivatives of the components f α of the section s. Now the term
r
n X
X
∂ 2f α j k
v w eα
j ∂xk
∂x
α=1
j,k=1
involving the second order partial derivatives of the components f α of the
section s is a symmetric function the vectors X and Y which remains invariant when the vectors X and Y are interchanged. Thus this term is eliminated
when we calculate DX (DY s) − DY (DX s). It follows from this that the function that sends X, Y and s to DX (DY s)−DY (DX s) is a first order differential
operator whose value at a point p of U is determined by the values of the
components v j , wk and f α and their first order partial derivatives at the
point p. Moreover the term
n
r
X
X
(Aα γ k ∂j f γ + Aα γ j ∂k f γ ) v j wk eα
j,k=1 α,γ=1
occurring in the expression for DX (DY s) also remains invariant when X and
Y are interchanged, and therefore is eliminated when we calculate DX (DY s)−
DY (DX s). We find that
DX (DY s) − DY (DX s)
r n X
k
k
X
j ∂v
j ∂w
−w
v
=
j
∂x
∂xj
j,k=1 α=1
+
r n
X
X
j,k=1 α,γ=1
r
X
α
r
∂f α X α
A γ kf γ
+
∂xk γ=1
!
eα
(∂j Aα γ k ) − (∂k Aα γ j )
(A β j A γ k − A β k A γ j ) v j wk f γ eα .
+
β
α
β
β=1
Now the term
n X
r k
k
X
j ∂w
j ∂v
v
−w
∂xj
∂xj
j,k=1 α=1
r
∂f α X α
+
A γ kf γ
∂xk γ=1
!
eα
is the covariant derivative of the section s with respect to a vector field on U
which, when expressed in terms of local coordinates x1 , x2 , . . . , xn takes the
form
r n X
k
k
X
∂
j ∂w
j ∂v
v
−
w
.
j
j
k
∂x
∂x
∂x
j,k=1 α=1
147
This vector field is the Lie bracket [X, Y ] of the vector fields X and Y . It
follows therefore that
DX (DY s) − DY (DX s) − D[X,Y ] s =
n
r
X
X
F α γ j k v j wk f γ eα ,
j,k=1 α,γ=1
where
r
F
α
γjk
∂Aα γ k ∂Aα γ j X α
=
−
+
(A β j Aβ γ k − Aα β k Aβ γ j ).
∂xj
∂xk
β=1
Now these quantities F α γ j k are the components of a smooth section of a vector bundle over M . This vector bundle is the tensor product E ⊗E ∗ ⊗T ∗ M ⊗
T ∗ M , where E ∗ is the dual bundle of E and T ∗ M is the cotangent bundle
of the smooth manifold M . Indeed there are smooth sections ε1 , ε2 , . . . , εr of
the dual bundle πE ∗ : E ∗ → M of E over the open set U characterized by the
property that hεα , eβ i = δβα at each point of U , where δβα is the Kronecker
delta, equal to 1 when α = β, but equal to zero otherwise. The values of the
smooth sections ε1 (p), ε2 (p), . . . , εr (p) at any point p of U constitute a basis
of the fibre Ep∗ of the dual bundle at p which is the dual basis corresponding
to the basis e1 (p), e2 (p), . . . , er (p) of Ep . Let
n
r
X
X
FD =
F α γ j k eα ⊗ εγ ⊗ dxj ⊗ dxk
j,k=1 α,γ=1
over the open set U . Then FD represents a smooth section of the smooth
vector bundle E ⊗ E ∗ ⊗ T ∗ M ⊗ T ∗ M over U . This section determines a
multilinear map
(FD )p : Ep × Tp M × Tp M → Tp M,
which sends (sp , Xp , Yp ) to FD (Xp , Yp )sp for all sp ∈ Ep and Xp , Yp ∈ Tp M ,
where
FD (Xp , Yp )sp =
n
r
X
X
F α γ j k eα hεγ , sp ihdxj , Xp i hdxk , Yp i.
j,k=1 α,γ=1
Thus if s is a section of πE : E → M over U , and if X and Y are vector fields
on U , where
X=
n
X
j=1
vj
∂
,
∂xj
Y =
n
X
wj
j=1
148
∂
∂xj
and s =
n
X
α=1
f α eα ,
then
FD (Xp , Yp )s(p) =
n
r
X
X
F α γ j k f γ v j wk eα .
j,k=1 α,γ=1
Our calculations thus show that there is a smooth section FD of the vector
bundle E ⊗ E ∗ ⊗ T ∗ M ⊗ T ∗ M characterized by the property that
DX (DY s) − DY (DX s) − D[X,Y ] s = FD (X, Y )s,
for all smooth sections of πE : E → M over the open set U , and for all smooth
vector fields X and Y on U . This section FD is the curvature of the smooth
connection D.
Let us consider in more detail the nature of sections of this vector bundle
E ⊗ E ∗ ⊗ T ∗ M ⊗ T ∗ M . Now E ⊗ E ∗ can be identified with the smooth vector
bundle End(E) whose fibre End(Ep ) at each point p of M is the algebra
of linear operators on the fibre Ep . (Linear operators on a vector space
are endomorphisms of that vector space.) Indeed elements of Ep ⊗ Ep∗ are
expressed as linear combinations of the form
r
X
S α γ eα (p) ⊗ εγ (p),
α,γ=1
with uniquely-determined real coefficients S α γ , where e1 (p), e2 (p), . . . , er (p) is
a basis of Ep and ε1 (p), ε2 (p), . . . , εr (p) is the corresponding dual basis of Ep∗ .
This linear combination of basis elements of Ep ⊗Ep∗ corresponds to the linear
r
r
P
P
operator that sends
cγ eγ (p) to
S α γ cγ eα (p) for all c1 , c2 , . . . , cr ∈ R.
α,γ=1
N
N2 ∗ γ=1
Tp M of the vector bundle 2 T ∗ M at each point p of M
The fibre
is a real vector space whose elements represent bilinear forms on the tangent
space Tp M at a point
as a direct sum of subV2 p.∗ This vector space splits
2 ∗
2 ∗
spaces S Tp M and
Tp M , where elements of S Tp M represent symmetric
V
bilinear forms on the tangent space Tp M , and where elements of 2 Tp∗ M represent skew-symmetric bilinear forms on this tangent space. Suppose that
the point p belongs to some open set U in M which is contained in the domain of a smooth coordinate system (x1 , x2 , . . . , xn ). Then the values at p of
the smooth tensor fields
dxj ⊗ dxk + dxk ⊗ dxj
for j ≤ k constitute at basis of the real vector space S 2 Tp∗ M . Similarly the
values at p of the smooth
fields dxj ∧ dxk for j < k constitute a basis
V2tensor
∗
of the real vector space
Tp M , where
dxj ∧ dxk = dxj ⊗ dxk − dxk ⊗ dxj .
149
It follows from this that the union of the vector spaces S 2 Tp∗ M for all points p
N
of M consitutes a smooth submanifold S 2 T ∗ M of 2 T ∗ M which is the total
space of a smooth vector bundle of rank 21 (n2 + n) over M (see ProposiV
tion 6.18). Similarly the union of the vector spaces 2 Tp∗ M for all points p
V
N
of M is a smooth submanifold 2 T ∗ M of 2 T ∗ M which is the total space
of a smooth vector bundle of rank 21 (n2 − n) over M (see Proposition 6.18).
V
Smooth sections of the vector bundle 2 T ∗ M are smooth differential forms
of degree two on the smooth manifold M .
The curvature FD of a smooth connection D on a smooth vector bundle
πE : E → M may therefore
V2 be∗ regarded as a smooth section of the
V2 smooth
vector bundle End(E) ⊗ T M over M whose fibre End(Ep ) ⊗ Tp∗ M at
each point p of M is a real vector space whose elements represent skewsymmetric bilinear maps from Tp M × Tp M to the space End(Ep ) of linear
operators on Ep .
We have shown the existence and basic
V2 ∗properties of the smooth section FD of the vector bundle End(E) ⊗ T M representing the curvature
of a smooth connection D on πE : E → M using calculations that involve
expressing smooth vector fields around a point p terms of local coordinates
(x1 , x2 , . . . , xn ) around p, and expressing sections of the smooth vector bundle πE : E → M as linear combinations of some chosen basis of sections
s1 , s2 , . . . , sr of the vector bundle around the point p. The existence and
basic properties of the curvature can be established by methods that do not
make explicit use of such local coordinate systems bases of local sections of
the vector bundle. Indeed we shall make use of Proposition 6.15 as a basic
tool to develop, in a more coordinate-free fashion, the theory of the curvature
of connections on smooth vector bundles over smooth manifolds.
Let Ẽ, E1 , E2 , . . . , Ek be smooth vector bundles over a smooth manifold M , and let Q be an operator that, over each open set U on M , assigns to
smooth sections s1 , s2 , . . . sk of the respective vector bundles E1 , E2 , . . . , Ek
defined over U a smooth section Q(s1 , s2 , . . . , sk ) of the vector bundle Ẽ
defined over this open set U . Suppose that this operator Q on sections is
R-multilinear, and that
Q(f1 s1 , f2 s2 , . . . , fk sk ) = f1 · f2 · · · fk Q(s1 , s2 , . . . , sk )
for all smooth functions f1 , f2 · · · fk on U , and for all s1 , s2 , . . . , sk , where sj
is a smooth section of the vector bundle Ej defined over U for j = 1, 2, . . . , k.
Proposition 6.15 then ensures that there exists a smooth section Q of the
vector bundle
Ẽ ⊗ E1∗ ⊗ E2∗ ⊗ · · · ⊗ Ek∗
150
such that
Q(s1 , s2 , . . . , sk ) = Q(s1 , s2 , . . . , sk )
for all s1 , s2 , . . . , sk , where sj is a smooth section of the vector bundle Ej over
U for j = 1, 2, . . . , k.
Proposition 8.4 Let D be a smooth connection on a smooth vector bundle
πE : E → M over a smooth manifold M , and let End(Ep ) denote the space
of linear operators on the fibre Ep of the vector bundle over the point p.
Given any smooth section s: U → X of this vector bundle, defined over some
open subset U of M , and given any smooth vector fields X and Y on U , let
FD (X, Y )s denote the smooth section of πE : E → M defined such that
FD (X, Y )s = DX (DY s) − DY (DX s) − D[X,Y ] s.
Then
FD (X, Y )(f s) = FD (f X, Y )s = FD (X, f Y )s = f FD (X, Y )s
for all smooth real-valued functions f on the open set U , and thus
V there
exists a smooth section FD of the smooth vector bundle End(E) ⊗ 2 T ∗ M
whose value at each point p of M represents a skew-symmetric bilinear map
(FD )p : Tp M × Tp M → End(Ep ) on the tangent space Tp M at each point p of
M which is defined such that
(FD )p (Xp , Yp )s(p) = (FD (X, Y )s)(p) = DXp (DY s) − DYp (DX s) − D[X,Y ]p s.
for all smooth vector fields X and Y defined around the point p and for all
smooth sections s of πE : E → M defined around p.
Proof Let s be a smooth section of the vector bundle πE : E → M defined
over some open set U in M , let X and Y be smooth vector fields on U , and
let f : U → R be a smooth real-valued function on U . Then
[X, Y ][f ] = X[Y [f ]] − Y [X[f ]],
[f X, Y ] = f [X, Y ] − Y [f ] X,
[X, f Y ] = f [X, Y ] + X[f ] Y
(see Lemma 7.6). It follows that
FD (f X, Y )s = Df X (DY s) − DY (Df X s) − D[f X,Y ] s
= f DX (DY s) − DY (f DX s) − Df [X,Y ]−Y [f ]X s
= f DX (DY s) − f DY (DX s) − Y [f ] DX s
151
=
=
FD (X, f Y )s =
=
FD (X, Y )(f s) =
=
=
=
=
− f D[X,Y ] s + Y [f ]DX s
f DX (DY s) − f DY (DX s) − f D[ X, Y ]s
f FD (X, Y )s,
−FD (f Y, X)s = −f FD (Y, X)s
f FD (X, Y )s,
DX (DY (f s)) − DY (DX (f s)] − D[X,Y ] (f s)
DX (Y [f ] s + f DY s) − DY (X[f ] s + f DY s)
− [X, Y ][f ] s − f D[X,Y ] s
Y [f ] DX s + X[Y [f ]] s + f DX (DY s) + X[f ] DY s
− X[f ] DY s − Y [X[f ]] s − f DY (DX s) − Y [f ] DX s
− [X, Y ][f ] s − f D[X,Y ] s
f (DX (DY s) − DY (DX s) − D[X,Y ] s)
+ X[Y ][f ] − Y [X[f ]] − [X, Y ][f ]
f FD (X, Y )s.
Moreover it is easy to see that
FD (X1 + X2 , Y )s = FD (X1 , Y )s + FD (X2 , Y )s,
FD (X, Y1 + Y2 )s = FD (X, Y1 )s + FD (X, Y2 )s,
FD (X, Y )(s1 + s2 ) = FD (X, Y )s1 + FD (X, Y )s2 .
for all sections s, s1 , s2 of the vector bundle and for all smooth vector fields
X, X1 , X2 , Y , Y1 and Y2 defined over the open set U of M .
It now follows from Proposition 6.15 that the V
operator FD determines a
smooth section FD of the vector bundle End(E) ⊗ 2 T ∗ M with the required
properties.
Definition Let D be a smooth connection defined on a smooth vector bundle
πE : E → M over a smooth manifold M . Let s: U → E be a smooth section
of this vector bundle, defined over some open set U in M , and let X and
Y be smooth vector fields on U . We define the curvature FD of the smooth
connection
D to be the smooth section of the smooth vector bundle End(E)⊗
V2 ∗
T M characterized by the property that
FD (X, Y )s = DX (DY s) − DY (DX s) − D[X,Y ] s.
for all smooth vector fields X and Y and smooth sections s of πE : E → M ,
where these vector fields and sections are all defined over some open set in
M.
152
The following proposition summarizes the results of calculations presented
above.
Proposition 8.5 Let πE : E → M be a smooth vector bundle of rank r over
a smooth manifold M , and, for each p ∈ M , let Ep denote the fibre of this
bundle over the point p. Let U be an open set in M , let (x1 , x2 , . . . , xn ) be
a smooth coordinate system for M defined over U , and let e1 , e2 , . . . , er be
smooth sections of πE : E → M over U , where, for each point p of U , the
elements e1 (p), e2 (p), . . . , er (p) constitute a basis of the real vector space Ep .
Let D be a smooth connection on this vector bundle, and let Aα β j be the
smooth real-valued functions on U defined such that
D
Then
FD =
∂
∂xj
eβ =
n
r
X
X
r
X
Aα β j eα ,
α=1
F α γ j k eα ⊗ εγ ⊗ dxj ⊗ dxk
j,k=1 α,γ=1
where
r
F
α
γjk
∂Aα γ k ∂Aα γ j X α
=
−
+
(A β j Aβ γ k − Aα β k Aβ γ j ).
∂xj
∂xk
β=1
Proposition 8.6 Let D be a smooth connection defined on a smooth vector
bundle πE : E → M over a smooth manifold M , and let FD be the curvature
of D. Let s: M → E be a smooth section of the vector bundle πE : E → M ,
and let X, Y and Z be smooth vector fields on M . Then
DX (FD (Y, Z)s) + DY (FD (Z, X)s) + DZ (FD (X, Y )s)
= FD ([Y, Z], X)s + FD ([Z, X], Y )s + FD ([X, Y ], Z)s
+ FD (Y, Z)(DX s) + FD (Z, X)(DY s) + FD (X, Y )(DZ s).
Proof It follows from Proposition 8.4 and the definition of connections on
smooth bundles that
DX (FD (Y, Z)s) + DY (FD (Z, X)s) + DZ (FD (X, Y )s)
= DX (DY (DZ s) − DZ (DY s) − (D[Y,Z] s))
+ DY (DZ (DX s) − DX (DZ s) − (D[Z,X] s))
+ DZ (DX (DY s) − DY (DX s) − (D[X,Y ] s))
= DX (DY (DZ s)) − DX (DZ (DY s)) − DX (D[Y,Z] s)
153
+ DY (DZ (DX s)) − DY (DX (DZ s)) − DY (D[Z,X] s)
+ DZ (DX (DY s)) − DZ (DY (DX s)) − DZ (D[X,Y ] s)
= FD (Y, Z)(DX s) + D[Y,Z] (DX s) − DX (D[Y,Z] s)
+ FD (Z, X)(DY s) + D[Z,X] (DY s) − DY (D[Z,X] s)
+ FD (X, Y )(DZ s) + D[Z,X] (DZ s) − DZ (D[X,Y ] s)
= FD (Y, Z)(DX s) + FD ([Y, Z], X)s + D[[Y,Z],X] s
+ FD (Z, X)(DY s) + FD ([Z, X], Y )s + D[[Z,X],Y ] s
+ FD (X, Y )(DZ s) + FD ([X, Y ], Z)s + D[[X,Y ],Z] s
= FD (Y, Z)(DX s) + FD (Z, X)(DY s) + FD (X, Y )(DZ s)
+ FD ([Y, Z], X)s + FD ([Z, X], Y )s + FD ([X, Y ], Z)s
+ D[[Y,Z],X]+[Z,X],Y ]+[[X,Y ],Z] s.
But the Lie Bracket satisfies the Jacobi Identity, and therefore
[[Y, Z], X] + [Z, X], Y ] + [[X, Y ], Z] = 0.
(see Lemma 7.5). The result follows.
8.3
Induced Connections on Dual Bundles
Proposition 8.7 Let D be a smooth connection on a smooth vector bundle
πE : E → M . Then D induces a connection (which we also denote by D)
on the dual bundle πE ∗ : E ∗ → M . This connection on the dual bundle is
defined such that if ϕ is a smooth section of πE ∗ : E ∗ → M defined around
some point p of M then
hDXp ϕ, s(p)i = Xp [hϕ, si] − hϕ, DXp si
for all smooth sections s of πE : E → M defined around the point p, and for
all tangent vectors Xp to M at the point p.
Proof Let X be a smooth vector field, let s be a smooth section of πE : E →
M , and let f be a smooth real-valued function defined throughout some open
set U in M . Then
X[hϕ, f si] − hϕ, DX (f s)i
= X[f · hϕ, si] − hϕ, f DX s + X[f ] si
= X[f ]hϕ, si + f X[hϕ, si]
− f hϕ, DX si − X[f ] hϕ, si
= f (X[hϕ, si] − hϕ, DX si) .
154
It follows from a direct application of Proposition 6.15 that there is a welldefined smooth section DX ϕ of πE ∗ : E ∗ → M characterized by the property
that
hDX ϕ, si = X[hϕ, si] − hϕ, DX si
for all smooth sections s of πE : E → M defined around the point p. Moreover this differential operator satisfies the properties required of a smooth
connection on the vector bundle πE ∗ : E ∗ → M .
8.4
Induced Connections on Tensor Products of Vector
Bundles
Proposition 8.8 Let E, E1 , . . . , Ek be smooth vector bundles over a smooth
manifold M , and let
M(E1 , E2 , . . . , Ek ; E)
denote the smooth vector bundle E ⊗ E1∗ ⊗ E2∗ ⊗ · · · ⊗ Ek∗ whose fibre at each
point p of M is the real vector space whose elements are multilinear maps
from (E1 )p × (E2 )p × · · · × (Ek )p to Ep . Let DE be a smooth connection
on the smooth vector bundle E, and let DEj be a smooth connection on the
smooth vector bundle Ej for j = 1, 2, . . . , k. Then these smooth connections
DE and DEj induce a smooth connection D on the smooth vector bundle
M(E1 , E2 , . . . , Ek ; E) characterized by the property that
(DXp S)(s1 , s2 , s3 , . . . , sk )
E
= DX
S(s
,
s
,
s
,
.
.
.
,
s
)
1 2 3
k
p
E2
E1
s , s3 , . . . , sk )
s , s2 , s3 , . . . , sk ) − S(s1 , DX
− S(DX
p 2
p 1
Ek
E2
− S(s1 , s2 , DX
s )
s , . . . , sk ) − · · · − S(s1 , s2 , s3 , . . . , DX
p k
p 3
for all s1 , s2 , . . . , sk , where sj is a smooth section of the smooth vector bundle Ej for j = 1, 2, . . . , k, and where these sections s1 , s2 , . . . , sk are all defined over some open set U in M .
Proof Let f be a smooth real-valued function defined on a neighbourhood
of a point p of M . Then
(Df Xp S)(s1 , s2 , s3 , . . . , sk ) = f (DXp S)(s1 , s2 , s3 , . . . , sk ).
Also
E
E
DX
S(f
s
,
s
,
s
,
.
.
.
,
s
)
=
D
f
S(s
,
s
,
s
,
.
.
.
,
s
)
1 2 3
k
1 2 3
k
Xp
p
= Xp [f ] S(s1 , s2 , s3 , . . . , sk )
E
+ f DXp S(s1 , s2 , s3 , . . . , sk )
155
and
E1
(f s1 ), s2 , s3 , . . . , sk ) = Xp [f ] S(s1 , s2 , s3 , . . . , sk )
S(DX
p
E1
s , s2 , s3 , . . . , sk ),
+ f S(DX
p 1
and therefore
(DXp S)(f s1 , s2 , s3 , . . . , sk ) = f (DXp S)(s1 , s2 , s3 , . . . , sk ).
Similarly
(DXp S)(s1 , f s2 , s3 , . . . , sk ) = f (DXp S)(s1 , s2 , s3 , . . . , sk ),
(DXp S)(s1 , s2 , f s3 , . . . , sk ) = f (DXp S)(s1 , s2 , s3 , . . . , sk ),
..
.
(DXp S)(s1 , s2 , s3 , . . . , f sk ) = f (DXp S)(s1 , s2 , s3 , . . . , sk ).
The required result follows immediately on applying Proposition 6.15.
Proposition 8.9 Let E1 , . . . , Ek be smooth vector bundles over a smooth
manifold M , and let DEj be a smooth connection on the smooth vector bundle Ej for j = 1, 2, . . . , k. Then these smooth connections Ej induce a smooth
connection D on the smooth vector bundle E1 ⊗ E2 ⊗ · · · ⊗ Ek characterized
by the property that
DXp (s1 ⊗ s2 ⊗ s3 ⊗ · · · ⊗ sk )
E2
E1
s ⊗ s3 ⊗ · · · ⊗ sk
s ⊗ s2 ⊗ s3 ⊗ · · · ⊗ sk + s1 ⊗ D X
= DX
p 2
p 1
Ek
E3
+ s1 ⊗ s2 ⊗ DX
s ⊗ · · · ⊗ sk + · · · + s1 ⊗ s2 ⊗ s3 ⊗ · · · ⊗ DX
s
p k
p 3
for all s1 , s2 , . . . , sk , where sj is a smooth section of the smooth vector bundle Ej for j = 1, 2, . . . , k, and where these sections s1 , s2 , . . . , sk are all defined over some open set U in M .
Proof At each point p of M the tensor product
(E1 )p ⊗ (E2 )p ⊗ · · · ⊗ (Ek )p
of the fibres of the vector bundles can be identified with the space
M((E1 )∗p , (E2 )∗p , . . . , (Ek )∗p ; R)
of multilinear maps from (E1∗ )p × (E2∗ )p × · · · × (Ek∗ )p to the field R of real
numbers, where Ej∗ is the dual bundle of Ej for j = 1, 2, . . . , k. Thus a smooth
156
section of the tensor product bundle E1 ⊗ E2 ⊗ · · · ⊗ Ek can be represented
as a function that assigns to each point p of M a multilinear map
Sp : (E1∗ )p × (E2∗ )p × · · · × (Ek∗ )p → R,
where the function S(ϕ1 , ϕ2 , . . . , ϕk ) sending p ∈ M to
Sp (ϕ1 (p), ϕ2 (p), . . . , ϕk (p))
is smooth for all ϕ1 , ϕ2 , . . . , ϕk , where ϕj is a smooth section of Ej∗ for j =
1, 2, . . . , k. Moreover if the operator S = s1 ⊗ s2 ⊗ · · · ⊗ sk , where sj is a
smooth section of the vector bundle Ej for j = 1, 2, . . . , k, then
S(ϕ1 , ϕ2 , . . . , ϕk ) = hϕ1 , s1 i · hϕ2 , s2 i · · · hϕk , sk i.
It follows from Proposition 8.8 that there is an induced connection D on
E1 ⊗ E2 ⊗ · · · ⊗ Ek , where
(DX (s1 ⊗ s2 ⊗ · · · ⊗ sk ))(ϕ1 , ϕ2 , . . . , ϕk )
= X[S(ϕ1 , ϕ2 , . . . , ϕk )]
E∗
E∗
− S(DX1 ϕ1 , ϕ2 , . . . , ϕk ) − S(ϕ1 , DX2 ϕ2 , . . . , ϕk )
E∗
− · · · − S(ϕ1 , ϕ2 , . . . , DXk ϕk )
But
X[S(ϕ1 , ϕ2 , . . . , ϕk )] = X[hϕ1 , s1 i · hϕ2 , s2 i · · · hϕk , sk i]
k
X
Y
hϕm , sm i
=
X[hϕj , sj i] ·
j=1
m6=j
and
E∗
∗
S(ϕ1 , . . . , ϕj−1 , DEj ϕj , ϕj+1 , . . . , ϕk ) = hDXj ϕj , sj i
Y
m6=j
and
E∗
E
X[hϕj , sj i] = hDXj ϕj , sj i + hϕj , DXj sj i
for j = 1, 2, . . . , k. It follows that
(DX (s1 ⊗ s2 ⊗ · · · ⊗ sk ))(ϕ1 , ϕ2 , . . . , ϕk )
k Y
X
E∗
=
X[hϕj , sj i] − hDXj ϕj , sj i
hϕm , sm i
j=1
m6=j
157
hϕm , sm i.
=
k
X
j=1
=
k
X
E
hϕj , DXj sj i
Y
hϕm , sm i
m6=j
E
(s1 ⊗ · · · ⊗ sj−1 ⊗ DXj sj ⊗ sj+1 ⊗ · · · ⊗ sk )(ϕ1 , ϕ2 , . . . , ϕk )
j=1
and thus
DX (s1 ⊗ s2 ⊗ · · · ⊗ sk )) =
k
X
E
s1 ⊗ · · · ⊗ sj−1 ⊗ DXj sj ⊗ sj+1 ⊗ · · · ⊗ sk ,
j=1
as required.
8.5
Affine Connections on Smooth Manifolds
Definition An affine connection ∇ on a smooth manifold M is a connection
on the tangent bundle πT M : T M → M of M .
Thus an affine connection ∇ on a smooth manifold M is a differential
operator which, at each point p of M , associates a tangent vector ∇Xp Y to
each smooth vector field Y defined around p and to each tangent vector Xp
at p, and which satisfies the following conditions:— of M :
(i) ∇Wp +Xp Y = ∇Wp Y + ∇Xp Y
for all tangent vectors Wp and Xp at a point p of M , and for all smooth
vector fields Y defined around p;
(iii) ∇c Xp Y = c ∇Xp Y
for all real numbers c and tangent vectors Xp at a point p of M , and
for all smooth vector fields Y defined around p;
(iii) ∇Xp (Y + Z) = ∇Xp Y + ∇Xp Z
for all tangent vectors Xp at a point p of M , and for all smooth vector
fields Y and Z defined around p;
(iv) ∇X (f Y ) = X[f ] Y + f ∇X Y
for all tangent vectors Xp at a point p of M , for all smooth real-valued
functions f defined around p, and for all smooth vector fields Y defined
around p;
(v) given smooth vector fields X and Y defined over a subset U of M , the
function that sends points p of U to ∇Xp Y is itself a smooth vector
field defined over U .
158
The tangent vector ∇Xp Y at a point p of the manifold determined by a
tangent vector Xp at p and a smooth vector field Y defined around p is
referred to as the covariant derivative of the vector field Y along the tangent
vector Xp (with respect to the affine connection ∇).
Let ∇ be an smooth affine connection on a smooth manifold M , and let
T be the R-bilinear operator acting on smooth vector fields on M defined
such that if U is an open set in M and if X and Y are smooth vector fields
on U , then
T (X, Y ) = ∇X Y − ∇Y X − [X, Y ].
If f is a smooth real-valued function and if X and Y are smooth vector fields
defined over an open set U in M then
[X, f Y ] = f [X, Y ] + X[f ] Y
(see Lemma 7.6). Therefore
T (X, f Y ) = ∇X (f Y ) − ∇f Y X − [X, f Y ]
= f ∇X Y + X[f ] Y − f ∇Y X − f [X, Y ] − X[f ] Y
= f T (X, Y ).
Also
T (f X, Y ) = −T (Y, f X) = −f T (Y, X) = f T (X, Y ).
It therefore follows from Proposition 6.15 that there is a smooth tensor field T
of type (1, 2) on M such that
T (X, Y ) = T (X, Y ) = ∇X Y − ∇Y X − [X, Y ]
for all smooth vector fields X and Y that are defined over some open subset
of M .
Also let R be the R-trilinear operator acting on smooth vector fields on
M defined such that if U is an open set in M and if X, Y and Z are smooth
vector fields on U , then
R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z
on U . It then follows from Proposition 8.4 that
R(f X, Y )Z = R(X, f Y )Z = R(X, Y )(f Z) = f R(X, Y )Z
for all smooth real-valued functions f and smooth vector fields X, Y , Z on
U , and thus there exists a tensor field R of type (1, 3) on M such that
R(X, Y )Z = R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z
for all smooth vector fields X, Y and Z that are defined over some open
subset of M .
159
Definition The torsion tensor T and the curvature tensor R of a smooth
affine connection ∇ are the smooth tensor fields of types (1, 2) and (1, 3)
respectively on M defined such that if U is an open subset of M , and if X,
Y and Z are smooth vector fields on U , then
T (X, Y ) = ∇X Y − ∇Y X − [X, Y ]
and
R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z.
An affine connection ∇ on M is said to be torsion-free if its torsion tensor is
everywhere zero (so that ∇X Y − ∇Y X = [X, Y ] for all smooth vector fields
X and Y on M ).
Note that the torsion tensor T and the curvature tensor R of a smooth
affine connection ∇ on a smooth manifold M satisfy T (X, Y ) = −T (Y, X)
and R(X, Y )Z = −R(Y, X)Z for all smooth vector fields X, Y and Z on M .
Example Let U be an open set in Rn , and let X and Y be smooth vector
fields on U . Then
X=
n
X
vi
i=1
∂
,
∂xi
Y=
n
X
i=1
wi
∂
,
∂xi
where v 1 , v 2 , . . . , v n and w1 , w2 , . . . , wn are the components of the vector fields
X and Y with respect to the Cartesian coordinate system (x1 , x2 , . . . , xn ) on
Rn . The directional derivative ∂X Y of the vector field Y along the vector
field X is then given by the formula
∂X Y =
n
X
i=1
X[wi ]
n
i
X
∂
j ∂w ∂
=
v
∂xi i,j=1 ∂xj ∂xi
(where X[wi ] denotes the directional derivative of the function wi along the
vector field X). Then the differential operator sending smooth vector fields X
and Y to ∂X Y is an affine connection on U . We refer to this affine connection
as the canonical (or usual ) flat connection on the open set U . Now
n X
n i
i
X
∂
j ∂w
j ∂v
∂X Y − ∂Y X =
v
−w
= [X, Y]
j
j
∂x
∂x ∂xi
i=1 j=1
(see Lemma 7.7). Thus the canonical flat connection ∂ on U is torsion-free.
Moreover, given any smooth vector field Z on U with Cartesian compoents
160
c1 , c2 , . . . , cn , we see that
∂X ∂Y Z − ∂Y ∂X Z = (∂X ∂Y − ∂Y ∂X )
n
X
i=1
=
n
X
i=1
∂
ci i
∂x
!
n
X
∂
∂
=
X[Y[c ]] − Y[X[c ]]
[X, Y][ci ] i
i
∂x
∂x
i=1
i
i
= ∂[X,Y] Z.
We deduce that the curvature tensor of the canonical flat connection ∂ on U
is zero everywhere on U .
Example Let M be a smooth n-dimensional submanifold of k-dimensional
Euclidean space. The Levi-Civita connection on M is the smooth connection ∇ on the tangent bundle πT M : T M → M that is defined such that
(∇X Y).Z = (∂X Y).Z for all tangential vector fields X, Y and Z on M ,
where the Cartesian components of ∂X Y at a point p of M are the directional derivatives of those of Y along the tangent vector Xp .
Let f be a smooth function defined on an open set U in Rk , and let
v 1 , v 2 , . . . , v k and w1 , w2 , . . . , wk be smooth real-valued functions on U that
at each point p of M ∩ U are the Cartesian components of the tangential
vectors Xp and Yp , so that
X = (v 1 , v 2 , . . . , v k ),
Y = (w1 , w2 , . . . , wk )
throughout M ∩ U . Then
Y[f ] =
k
X
j=1
wj
∂f
∂xj
on M ∩ U , where x1 , x2 , . . . , xk are the standard Cartesian coordinate functions on Rk , and therefore
k
X
∂
j ∂f
X[Y[f ]] =
v
w
∂xi
∂xj
i,j=1
k j
2
X
i ∂w ∂f
i j ∂ f
v
+v w
=
∂xi ∂xj
∂xi ∂xj
i,j=1
i
and therefore
[X, Y][f ] = X[Y[f ]] − Y[X[f ]]
161
k X
k j
j
X
∂f
i ∂v
i ∂w
=
−w
v
i
i
∂x
∂x ∂xj
j=1 i=1
k
X
=
j=1
∂f
X[wj ] − Y[v j ]
∂xj
where [X, Y] is the Lie bracket of the smooth tangential vector fields X and
Y. It follows that [X, Y] is the tangential vector field whose jth Cartesian
component on M ∩ U is the smooth real-valued function X[wj ] − Y[v j ]. Thus
[X, Y] = ∂X Y − ∂Y X.
On taking the orthogonal projection of both sides of this equation onto the
tangent space at each point of the submanifold M , we find that
[X, Y] = ∇X Y − ∇Y X.
Thus the Levi-Civita connection ∇ on a smooth submanifold M of some
Euclidean space Rk is torsion-free.
Lemma 8.10 Let ∇ be an affine connection on a smooth manifold M , let
(x1 , x2 , . . . , xn ) be a smooth smooth coordinate system defined over an open
set U in M . Let X and Y be smooth vector fields on U , and let v 1 , v 2 , . . . , v n
and w1 , w2 , . . . , wn be the components of the vector fields X and Y with respect to the smooth coordinate system, so that
X=
n
X
vi
i=1
Then
∇X Y =
n
X
i,j=1
∂
,
∂xi
Y =
n
X
wi
i=1
∂wi
vj j
∂x
+
n
X
∂
.
∂xi
!
j
v w
k
Γijk
k=1
∂
,
∂xi
on U where the coefficients Γijk are smooth functions defined over U so that
n
∂ X
∂
=
∇ ∂
Γijk i ,
k
∂x
∂x
i=1
∂xj
for j, k = 1, 2, . . . , n.
Proof This is a special case of Proposition 8.1 and can be verified directly
by a straightforward computation.
162
Proposition 8.11 Let ∇ be a smooth affine connection on a smooth manifold M , and let T and R be the torsion and curvature tensors respectively of
∇. Let (x1 , x2 , . . . , xn ) be a smooth coordinate system for M defined over an
open subset U of M , and let
n
∂ X
∂
∇ ∂
=
Γljk l ,
k
∂x
∂x
l=1
∂xj
on U . Then
n
X
∂
T =
(Γljk − Γlkj ) l ⊗ dxj ⊗ dxk ,
∂x
l=1
and
R=
n
X
Rl ijk
l=1
∂
⊗ dxi ⊗ dxj ⊗ dxk
l
∂x
on U where
n
∂Γlki ∂Γlji X l m
=
−
+
(Γjm Γki − Γlkm Γm
ji ).
j
k
∂x
∂x
m=1
l
R ijk
Moreover if X, Y and Z are smooth vector fields over U , and if
X=
n
X
j=1
uj
∂
,
∂xj
Y =
then
T (X, Y ) =
n
X
vk
k=1
n
X
∂
,
∂xk
Z=
(Γljk − Γlkj )uj v k
j,k,l=1
and
R(X, Y )Z =
n
X
i,j,k,l=1
Rl ijk wi uj v k
n
X
i=1
wi
∂
,
∂xi
∂
∂xl
∂
.
∂xl
Proof The formula for the components of the torsion tensor follows directly
from the definition of that tensor. The formula for the coefficients Rm ijk of
the curvature tensor is a special case of the formula for the curvature of a
smooth connection stated in Proposition 8.5. In order to verify it directly,
we note that
∂
∂
=0
,
∂xj ∂xk
163
(see Corollary 7.8). Therefore
∂
∂
∂
∂
∂
R
, k
= ∇ ∂j ∇ ∂
− ∇ ∂j ∇ ∂
j
i
i
∂x
∂x
∂xk ∂x
∂xk ∂xi
∂x ∂x
∂x
n X
l ∂
l ∂
=
∇ ∂ j Γki l − ∇ ∂ Γji l
∂x
∂xk
∂x
∂x
l=1
!
n
X
∂
∂Γlki ∂Γlji
=
−
j
k
∂x
∂x
∂xl
l=1
n X
∂
∂
l
l
+
Γki ∇ ∂ j l − Γji ∇ ∂
∂x ∂x
∂xk ∂xl
l=1
!
n
X
∂Γlki ∂Γlji
∂
=
−
j
k
∂x
∂x
∂xl
l=1
+
n
X
l
m l
Γm
jl Γki − Γkl Γji
l,m=1
=
n
X
l=1
∂
∂xm
n
∂Γlki ∂Γlji X l m
l
m
−
+
Γ
Γ
−
Γ
Γ
jm
ki
km
ji
∂xj
∂xk m=1
!
∂
,
∂xl
as required.
8.6
Covariant Derivatives of Tensor Fields
The dual of the tangent bundle πT M : T M → M of a smooth manifold is
the cotangent bundle πT ∗M : T ∗ M → M . Smooth sections of this cotangent
bundle represent smooth 1-forms on the smooth manifold M . It follows
from Proposition 8.7 that an affine connection on M induces a connection
on the cotangent bundle of M . The following lemma summarizes the basis
properties of this connection.
Lemma 8.12 Let ∇ be a smooth affine connection on a smooth manifold M .
Then the affine connection ∇ on the tangent bundle πT M : T M → M induces
a smooth connection on the cotangent bundle πT ∗ M : T ∗ M → M . This connection is defined so that if U is an open set in M , if X and Y are smooth
vector fields defined over U , and if ϕ is a smooth 1-form on U , then
X[hϕ, Y i] = h∇X ϕ, Y i + hϕ, ∇X Y i.
If (x1 , x2 , . . . , xn ) is a smooth coordinate system defined over an open set U
164
in M , and if
n
∂ X
∂
l
∇ ∂
=
Γ
jk
∂xk
∂xl
l=1
j
∂x
for j, k = 1, 2, . . . , n, then
∇
∂
∂xj
l
dx = −
n
X
Γljk dxk .
k=1
Proof The existence and basic properties of the connection on the cotangent
bundle of M induced by ∇ follow on applying Proposition 8.7. It follows from
the definition of this induced connection that
∂
∂
∂
∂
l
l
l
∇ ∂ j dx , k
=
dx , k − dx , ∇ ∂ j k
∂x
∂x ∂x
∂x
∂xj
∂x
*
+
n
X
∂
∂ l
Γm
(δ ) − dxl ,
=
jk
∂xj k
∂xm
m=1
= −Γljk
where δkl is the Kronecker delta, equal to 1 when l = k, and equal to zero
otherwise. Thus
n
X
l
Γljk dxk ,
∇ ∂ j dx = −
∂x
k=1
as required.
The following proposition establishes the standard formula for the covariant derivative of a tensor field of type (r, s) on a smooth manifold, when
expressed in terms of components with respect to a local coordinate system.
Proposition 8.13 Let ∇ be a smooth affine connection on a smooth manifold M , and let (x1 , x2 , . . . ,n ) be a smooth local coordinate system defined
over an open subset U of M . Let S be a smooth tensor field of type (r, s)
defined over U , where
S =
n
X
n
X
∂
∂
∂
r
⊗
⊗
·
·
·
⊗
Skj11,j,k22,...,j
,...,ks
∂xj1
∂xj2
∂xjr
j1 ,j2 ,...,jr =1 k1 ,k2 ,...,kr =1
⊗ dxk1 ⊗ dxk2 ⊗ · · · ⊗ dxks
165
r
where the components Skj11,j,k22,...,j
,...,ks of S are smooth real-valued functions on
U . Then the covariant derivative of S with respect to the induced connection on the smooth vector bundle over M whose sections are tensor fields of
type (r, s) is determined with respect to the smooth local coordinate system
(x1 , x2 , . . . , xn ) by the following formula:—
∇
∂
∂xm
n
X
n
X
∂
∂
∂
⊗ j2 ⊗ · · · ⊗ jr
j
1
∂x
∂x
∂x
j1 ,j2 ,...,jr =1 k1 ,k2 ,...,kr =1
⊗ dxk1 ⊗ dxk2 ⊗ · · · ⊗ dxks ,
S =
r
Skj11,j,k22,...,j
,...,ks ;m
where
r
∂Skj11,j,k22,...,j
,...,ks
=
∂xm
n
X j l,j ,...,j
j2 j1 ,l,...,jr
jr j1 ,j2 ,...,l
r
1
2
Γml
+
Γ
+
·
·
·
+
Γ
S
+
Sk1 ,k
S
ml k1 ,k2 ,...,ks
ml k1 ,k2 ,...,ks
2 ,...,ks
r
Skj11,j,k22,...,j
,...,ks ;m
l=1
−
n
X
j1 ,j2 ,...,jr
j1 ,j2 ,...,jr
j1 ,j2 ,...,jr
l
l
Γlmk1 Sl,k
+
Γ
+
·
·
·
+
Γ
S
S
mk2 k1 ,l,...,ks
mks k1 ,k2 ,...,l .
2 ,...,ks
l=1
Proof We note that
S=
n
X
n
X
k1 ,k2 ,...,ks
r
,
Skj11,j,k22,...,j
,...,ks Pj1 ,j2 ,...,jr ⊗ Q
j1 ,j2 ,...,jr =1 k1 ,k2 ,...,kr =1
where
∂
∂
∂
⊗
⊗
·
·
·
⊗
j
j
∂x 1
∂x 2
∂xjr
= dxk1 ⊗ dxk2 ⊗ · · · ⊗ dxks
Pj1 ,j2 ,...,jr =
Qk1 ,k2 ,...,ks
for j1 , j2 , . . . , jr , k1 , k2 , . . . , ks = 1, 2, . . . , n. Then
∂
∂
∂
∇ ∂m j1 ⊗ j2 ⊗ · · · ⊗ jr
∇ ∂m Pj1 ,j2 ,...,jr =
∂x
∂x
∂x
∂x
∂x
∂
∂
∂
+ j1 ⊗ ∇ ∂m j2 ⊗ · · · ⊗ jr
∂x
∂x
∂x
∂x
∂
∂
∂
+ · · · + j1 ⊗ j2 ⊗ · · · ⊗ ∇ ∂m j2
∂x
∂x
∂x
∂x
166
=
n X
∂
∂
∂
⊗ j2 ⊗ · · · ⊗ jr
l
∂x
∂x
∂x
Γlmj1
l=1
∂
∂
∂
⊗
⊗
·
·
·
⊗
∂xj1
∂xl
∂xjr
∂
∂
∂ l
+ · · · + Γmjr j1 ⊗ j2 ⊗ · · · ⊗ l
∂x
∂x
∂x
n X
=
Γlmj1 Pl,j2 ,...,jr + Γlmj2 Pj1 ,l,...,jr
+ Γlmj2
l=1
+ · · · + Γlmjr Pj1 ,j2 ,...,l .
and
∇
∂
∂xm
k1 ,k2 ,...,ks
Q
=
k1
⊗ dxk2 ⊗ · · · ⊗ dxks
+ dxk1 ⊗ ∇ ∂m dxk2 ⊗ · · · ⊗ dxks
∂x
k1
+ · · · + dx ⊗ dxk2 ⊗ · · · ⊗ ∇ ∂m dxks
∇
∂
∂xm
dx
∂x
= −
n X
Γkml1 dxl ⊗ dxk2 ⊗ · · · ⊗ dxks
l=1
+ Γkml2 dxk1 ⊗ dxl ⊗ · · · ⊗ dxks
+ · · · + Γkmls dxk1 ⊗ dxk2 ⊗ · · · ⊗ dxl
= −
n X
Γkml1 Ql,k2 ,...,ks + Γkml2 Qk1 ,l,...,ks
l=1
+ · · · + Γkmls Qk1 ,k2 ,...,l .
Therefore
∇
∂
∂xm
S =
n
X
n
X
∂S j1 ,j2 ,...,jr
j1 ,j2 ,...,jr =1 k1 ,k2 ,...,kr =1
r
+ Skj11,j,k22,...,j
,...,ks ∇
∂
∂xm
k1 ,k2 ,...,ks
∂xm
Pj1 ,j2 ,...,jr ⊗ Qk1 ,k2 ,...,ks
r
+ Skj11,j,k22,...,j
,...,ks Pj1 ,j2 ,...,jr ⊗ ∇
=
n
X
n
X
∂
∂xm
∂S j1 ,j2 ,...,jr
j1 ,j2 ,...,jr =1 k1 ,k2 ,...,kr =1
Pj1 ,j2 ,...,jr ⊗ Qk1 ,k2 ,...,ks
k1 ,k2 ,...,ks
∂xm
167
Qk1 ,k2 ,...,ks
Pj1 ,j2 ,...,jr ⊗ Qk1 ,k2 ,...,ks
+
n
X
r
Skj11,j,k22,...,j
,...,ks
Γlmj1 Pl,j2 ,...,jr + Γlmj2 Pj1 ,l,...,jr
l=1
+ · · · + Γlmjr Pj1 ,j2 ,...,l ⊗ Qk1 ,k2 ,...,ks
−
n
X
r
Skj11,j,k22,...,j
,...,ks
Pj1 ,j2 ,...,jr ⊗ Γkml1 Ql,k2 ,...,ks + Γkml2 Qk1 ,l,...,ks
l=1
+ · · · + Γkmls Qk1 ,k2 ,...,l
,
Now it follows on relabelling indices of summation that
n
X
l
r
Skj11,j,k22,...,j
,...,ks Γmj1 Pl,j2 ,...,jr
j1 ,j2 ,...,jr ,l=1
n
X
=
l
r
Skj11,j,k22,...,j
,...,ks Γmj2 Pj1 ,l,...,jr =
n
X
j1 ,j2 ,...,jr ,l=1
n
X
2 ,...,jr
Γj1 P
Skl,j1 ,k
2 ,...,ks ml j1 ,j2 ,...,jr
l
r
Skj11,l,...,j
,k2 ,...,ks Γmj2 Pj1 ,j2 ,...,jr
j1 ,j2 ,...,jr ,l=1
j1 ,j2 ,...,jr ,l=1
etc. Therefore
∇
∂
∂xm
S =
n
X
n
X
k1 ,k2 ,...,ks
r
Skj11,j,k22,...,j
,...,ks ;m Pj1 ,j2 ,...,jr ⊗ Q
j1 ,j2 ,...,jr =1 k1 ,k2 ,...,kr =1
r
where Skj11,j,k22,...,j
,...,ks ;m is defined as in the statement of the proposition.
Example Let ∇ be a smooth affine connection on a smooth manifold M ,
and let (x1 , x2 , . . . ,n ) be a smooth local coordinate system defined over an
open subset U of M . Let H be a smooth tensor field of type (0, 2) defined
over U , where
n
X
H=
Hjk dxj ⊗ dxk .
j,k=1
Then
∇
∂
∂xm
H=
n
X
Hjk;m dxj ⊗ dxk ,
j,k=1
where
n
Hjk;m
∂Hjk X l
=
−
Γmj Hlk + Γlmk Hjl .
m
∂x
l=1
We can verify this identity using the basic method employed in the proof of
Proposition 8.13. Now smooth tensor fields of type (0, 2) are by definition
168
smooth sections of the vector bundle T ∗ M ⊗ T ∗ M . The induced connection
on this bundle is defined as described in the statement of Proposition 8.9. It
follows that
n
X
∂Hjk
∇ ∂m H =
dxj ⊗ dxk
m
∂x
∂x
j,k=1
n
X
+
Hjk
∇
∂
∂xm
dxj ⊗ dxk + dxj ⊗ ∇
∂
∂xm
dxk
j,k=1
But it follows from Lemma 8.12 that
∇
j
∂ dx = −
m
n
X
∂x
Γjmq dxq .
q=1
Therefore
n
X
∂Hjk
∇ ∂m H =
dxj ⊗ dxk
m
∂x
∂x
j,k=1
−
n X
Γjml Hjk
l
k
dx ⊗ dx +
Γkml Hjk
j
l
dx ⊗ dx
j,k,l=1
n
X
∂Hjk
dxj ⊗ dxk
=
m
∂x
j,k=1
−
n X
Γlmj Hlk dxj ⊗ dxk + Γlmk Hjl dxj ⊗ dxk
j,k,l=1
=
=
n
X
j,k=1
n
X
n
∂Hjk X l
l
Γ
H
+
Γ
H
−
lk
jl
mj
mk
∂xm
l=1
!
dxj ⊗ dxk
Hjk;m dxj ⊗ dxk .
j,k=1
Example Let ∇ be a smooth affine connection on a smooth manifold M ,
and let (x1 , x2 , . . . ,n ) be a smooth local coordinate system defined over an
open subset U of M . Let W be a smooth tensor field of type (1, 3) defined
over U , where
W =
n
X
i,j,k,l=1
W l ijk
∂
⊗ dxi ⊗ dxj ⊗ dxk .
∂xl
169
Then
∇
∂
∂xm
n
X
W =
W l ijk;m
i,j,k,l=1
∂
⊗ dxi ⊗ dxj ⊗ dxk ,
∂xl
where
n
l
Wijk;m
∂W l ijk X l
=
+
Γmq W q ijk − Γqmi W l qjk − Γqmj W l iqk − Γqmk W l ijq .
m
∂x
q=1
We can verify this identity using the basic method employed in the proof
of Proposition 8.13. Now smooth tensor fields of type (1, 3) are by definition
smooth sections of the vector bundle T M ⊗ T ∗ M ⊗ T ∗ M ⊗ T ∗ M . The
induced connection on this bundle is defined as described in the statement
of Proposition 8.9. It follows that
∇X W =
n
X
X[W l ijk ]
i,j,k,l=1
n
X
∂
⊗ dxi ⊗ dxj ⊗ dxk
l
∂x
∂ W ijk ∇X l ⊗ dxi ⊗ dxj ⊗ dxk
+
∂x
i,j,k,l=1
∂
+ l ⊗ ∇X dxi ⊗ dxj ⊗ dxk
∂x
∂
+ l ⊗ dxi ⊗ ∇X dxj ⊗ dxk
∂x
∂
+ l ⊗ dxi ⊗ dxj ⊗ ∇X dxk
∂x
l
for all smooth vector fields X on M , and therefore
n
X
∂W l ijk ∂
∇ ∂m W =
⊗ dxi ⊗ dxj ⊗ dxk
m
l
∂x
∂x
∂x
i,j,k,l=1
+
n
X
Γqml W l ijk
i,j,k,l,q=1
− Γimq W l ijk
− Γjmq W l ijk
− Γkmq W l ijk
∂
⊗ dxi ⊗ dxj ⊗ dxk
∂xq
∂
⊗ dxq ⊗ dxj ⊗ dxk
l
∂x
∂
⊗ dxi ⊗ dxq ⊗ dxk
l
∂x
∂
i
j
q
⊗ dx ⊗ dx ⊗ dx
∂xl
170
n
X
∂W l ijk ∂
=
⊗ dxi ⊗ dxj ⊗ dxk
m
l
∂x
∂x
i,j,k,l=1
n
X
+
Γlmq W q ijk
i,j,k,l,q=1
− Γqmi W l qjk
− Γqmj W l iqk
− Γqmk W l ijq
=
n
X
i,j,k,l=1
W l ijk;m
∂
⊗ dxi ⊗ dxj ⊗ dxk
l
∂x
∂
⊗ dxi ⊗ dxj ⊗ dxk
∂xl
∂
⊗ dxi ⊗ dxj ⊗ dxk
∂xl
∂
i
j
k
⊗
dx
⊗
dx
⊗
dx
∂xl
∂
⊗ dxi ⊗ dxj ⊗ dxk .
∂xl
This formula for the covariant derivative of a smooth tensor field of type
(1, 3) can also be established by applying Proposition 8.8.
8.7
The First Bianchi Identity
Proposition 8.14 (First Bianchi Identity) Let ∇ be a torsion-free affine
connection on a smooth manifold M . Let R denote the curvature operator of
∇. Then
R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0
Proof The connection ∇ is torsion-free, hence ∇X Y − ∇Y X = [X, Y ] for all
vector fields X and Y on M . Therefore
R(X, Y )Z + R(Y, Z)X + R(Z, X)Y
= ∇X ∇Y Z + ∇Y ∇Z X + ∇Z ∇X Y
− ∇Y ∇X Z − ∇Z ∇Y X − ∇X ∇Z Y
− ∇[X,Y ] Z − ∇[Y,Z] X − ∇[Z,X] Y
= ∇X (∇Y Z − ∇Z Y ) + ∇Y (∇Z X − ∇X Z)
+ ∇Z (∇X Y − ∇Y Z)
− ∇[X,Y ] Z − ∇[Y,Z] X − ∇[Z,X] Y
= ∇X [Y, Z] + ∇Y [Z, X] + ∇Z [X, Y ]
− ∇[X,Y ] Z − ∇[Y,Z] X − ∇[Z,X] Y
= 0.
171
Let (x1 , x2 , . . . , xn ) be a smooth coordinate system defined over an open
set U in M , and let the smooth real-valued functions (Rl ijk ) be defined such
that
n
X
∂
∂
∂
∂
R
, k
=
Rl ijk l .
j
i
∂x ∂x
∂x
∂x
l=1
The First Bianchi Identity then ensures that
Rl ijk + Rl jki + Rk kij = 0.
8.8
The Second Bianchi Identity
Let D be a smooth connection on a smooth principal bundle πE : E → M .
Then the connection D on E and the corresponding connection on the dual
bundle πE ∗ : E ∗ → M of E induce a smooth connection on the bundle End(E)
whose fibre over a point p of M is the space of linear operators on the
fibre Ep of the vector bundle E (see Proposition 8.8 and Proposition 8.9).
This connection on End(E) is defined such that
(DX K)(s) = DX (K(s)) − K(DX s)
for all smooth sections K of End(E), smooth sections s of E and smooth
vector fields X on M . It then follows from Proposition 8.8, or Proposition 8.9), that if ∇ is an affine connection on M , then the connection on
End(E) and the affine
V connection ∇ on M induce a connection on the vector
bundle End(E) ⊗ 2 T ∗ M , where
(DX Q)(Y, Z)(s) = DX (Q(Y, Z)s) − Q(∇X Y, Z)s − Q(Y, ∇X Z)s
− Q(X, Z)(DX s)
for all smooth sections s of πE : E → M and for all smooth vector fields X,
Y and Z on M .
Proposition 8.15 Let D be a smooth connection defined on a smooth vector
bundle πE : E → M over a smooth manifold M , let FD be the curvature of D,
and let ∇ be a torsion-free affine connection on M . Then
(DX FD )(Y, Z)s + (DY FD )(Z, X)s + (DZ FD )(X, Y )s = 0
for all smooth sections s of πE : E → M and smooth vector fields X, Y and
Z on M , where
(DX FD )(Y, Z)s = DX (FD (Y, Z)s) − FD (∇X Y, Z)s − FD (Y, ∇X Z)s
− FD (X, Y )(DX s).
172
Proof It follows from Proposition 8.6 that
DX (FD (Y, Z)s) + DY (FD (Z, X)s) + DZ (FD (X, Y )s)
= FD ([Y, Z], X)s + FD ([X, Z], Y )s + FD ([X, Y ], Z)s
+ FD (Y, Z)(DX s) + FD (Z, X)(DY s) + FD (X, Y )(DZ s).
But [X, Y ] = ∇X Y − ∇Y X, etc. because the affine connection ∇ is torsionfree. Also FD (X, Y )s = −FD (Y, X)s. It follows that
DX (FD (Y, Z)s) + DY (FD (Z, X)s) + DZ (FD (X, Y )s)
= FD (∇Y Z, X)s − FD (∇Z Y, X)s
+ FD (∇Z X, Y )s − FD (∇X Z, Y )s
+ FD (∇X Y, Z)s − FD (∇Y X, Z)s
+ FD (Y, Z)(DX s) + FD (Z, X)(DY s) + FD (X, Y )(DZ s)
= FD (∇X Y, Z)s + FD (Y, ∇X Z)s + FD (Y, Z)(DX s)
+ FD (∇Y Z, X)s + FD (Z, ∇Y X)s + FD (Z, X)(DY s)
+ FD (∇Z X, Y )s + FD (X, ∇Z Y )s + FD (X, Y )(DZ s),
and thus
(DX FD )(Y, Z)s + (DY FD )(Z, X)s + (DZ FD )(X, Y )s = 0,
as required.
We can apply Proposition 8.15 in the special case when the vector bundle E over M is the tangent bundle M , and where the smooth connection
is a torsion-free affine connection on M . That proposition then yields the
following result.
Corollary 8.16 (Second Bianchi Identity) Let ∇ be a torsion-free affine
connection on a smooth manifold M . Then
(∇X R)(Y, Z)W + (∇Y R)(Z, X)W + (∇Z R)(X, Y )W = 0
for all smooth vector fields X, Y , Z and W on M , where R is the curvature
tensor of the connection ∇, and where
(∇X R)(Y, Z)W = ∇X (R(Y, Z)W ) − R(∇X Y, Z)W − R(Y, ∇X Z)W
− R(X, Y )(∇X W ).
173
9
9.1
Riemannian and Pseudo-Riemannian Manifolds
Riemannian and Pseudo-Riemannian Metrics
Definition Let M be a smooth manifold. A metric tensor g on M is a tensor
field on M that assigns to each point p of M a non-degenerate symmetric
bilinear form gp on the tangent space Tp M to M at p.
Lemma 9.1 Let M be a smooth manifold, and let g be a metric tensor on
M . Then
g(Xp + Yp , Zp )
g(Xp , Yp + Zp )
g(c Xp , Yp )
g(Xp , Yp )
=
=
=
=
g(Xp , Zp ) + g(Yp , Zp ),
g(Xp , Yp ) + g(Xp , Zp ),
g(Xp , cYp ) = cg(Xp , Yp ),
g(Yp , Xp )
for all p ∈ M and Xp , Yp , Zp ∈ Tp M , and for all real numbers c. Moreover,
given any non-zero tangent vector Xp at some point p of M , there exists some
tangent vector Yp at p such that g(Xp , Yp ) 6= 0. Also, given any element θp
of the cotangent space Tp∗ M of M at p, there exists some tangent vector θp]
which satisfies
g(θp] , Yp ) = θp (Yp ) = hθp , Yp i
for all Yp ∈ Tp M .
Proof The given identities represent the fact that the metric tensor is a
symmetric bilinear form on each tangent space. The definition of nondegeneracy for bilinear forms on a vector space requires that, given any tangent vector X ∈ Tp M , there exists some tangent vector Y ∈ Tp M such that
g(Xp , Yp ) 6= 0. Let λp : Tp M → Tp∗ M be the linear transformation defined
such that λp (Xp ) = Xp[ , where
hXp[ , Yp i = gp (Xp , Yp )
for all Yp ∈ Tp M . Then the non-degeneracy of the bilinear form gp ensures
that the linear transformation λp is injective. But the domain Tp M and
codomain Tp∗ M of this linear transformation have the same dimension. It
follows from basic linear algebra that λp : Tp M → Tp∗ M is an isomorphism of
∗
vector spaces. Let θp] = λ−1
p (θp ) for all θp ∈ T M . Then
hθp , Yp i = hλp (θp] ), Yp , i = gp (θp] , Yp )
for all Yp ∈ Tp M , as required.
174
Definition A Riemannian metric on a smooth manifold M is a smooth
metric tensor that is positive-definite at each point of M .
Definition A pseudo-Riemannian metric on a smooth manifold M is a
smooth metric tensor that is nondegenerate but that need not be positivedefinite at each point of M .
Definition Let M be a Riemannian or pseudo-Riemannian manifold with
metric tensor g. The raising and lowering operators at a point p of M are
the isomorphisms ρ: Tp∗ M → Tp M and λ: Tp M → Tp∗ M between the tangent
and cotangent spaces of M at the point p defined such that ρ(θp ) = θp] and
λ(Xp ) = Xp[ for all θp ∈ Tp∗ M and Xp ∈ Tp M , where
g(θp] , Yp ) = hθp , Yp i and hXp[ , Yp ) = g(Xp , Yp ).
Let M be a Riemannian or pseudo-Riemannian manifold of dimension n,
with metric tensor g, and let (x1 , x2 , . . . , xn ) be a smooth coordinate system
defined over some open set U in M . Let
∂
∂
gij = g
,
∂xi ∂xj
for i, j = 1, 2, . . . , n. Then the components gij of the metric tensor g are
smooth real-valued functions on U . If M is a Riemannian manifold then the
values of these components at each point of U are the entries of a positivedefinite symmetric matrix. If M is a pseudo-Riemannian manifold then the
values of these components at each point of U are the entries of a symmetric
matrix that is non-singular but need not be positive definite.
Let X and Y are smooth vector fields defined over the open set U . Then
X=
n
X
i=1
∂
,
v
∂xi
i
Y =
n
X
i=1
wi
∂
,
∂xi
where v 1 , a2 , . . . , an and w1 , b2 , . . . , bn are smooth functions on U , and therefore
n
X
g(X, Y ) =
gij v i wj .
i,j=1
Now, at each point p of M , the matrix with entry gij in the ith row and
jth column is invertible. The inverse of this matrix has entries g kl , where
g kl = g lk and
n
X
gij g jk = δik ,
j=1
175
where δik denotes the Kronecker delta that has the value 1 when i = k, but
has the value 0 otherwise.
Lemma 9.2 Let M be a Riemannian or pseudo-Riemannian manifold with
metric tensor g, and let (x1 , x2 , . . . , xn ) be a smooth coordinate system for M
defined over an open subset U of M , let
∂
∂
gij = g
,
∂xi ∂xj
for i, j = 1, 2, . . . , n, and let g kl be the smooth functions defined on U so
n
P
that g kl = g lk and
gij g jk = δik , where δik denotes the Kronecker delta. Let
j=1
Xp ∈ Tp M and θp ∈ Tp∗ M be elements of the tangent and cotangent spaces
to M at some point p of M , and let Xp[ and θp] be defined such that
g(θp] , Yp ) = hθp , Yp i
Let
n
X
∂ Xp =
a
∂xj p
j=1
Then
Xp[
=
n
X
j
gjk a
j
dxkp
hXp[ , Yp ) = g(Xp , Yp ).
and
and
θp =
ck dxkp .
k=1
θp]
and
j,k=1
Proof Let
n
X
n
X
n
X
∂ g ck
.
=
∂xj p
j,k=1
jk
∂ Yp =
b
.
k
∂x
p
k=1
k
Then
*
n
X
+
gjk a
j
dxkp ,
Yp
n
X
=
gjk aj bk = g(Xp , Yp ) = hXp[ , Yp i.
j,k=1
j,k=1
This identity holds for all Y ∈ Tp M , and therefore
n
X
gjk aj dxkp = Xp[
j,k=1
Also
!
∂ g
g ck
, Yp
=
∂xj p
j,k=1
n
X
jk
=
n
X
i,j,k=1
n
X
jk
i
g ck b gji =
n
X
δik ck bi
i,k=1
ck bk = hθp , Yp i = g(θp] , Yp ).
k=1
176
This identity holds for all Y ∈ Tp M , and therefore
n
X
∂ jk
g ck
= θp] ,
j
∂x
p
j,k=1
as required.
Let M be a Riemannian or pseudo-Riemannian manifold with metric
tensor g, let (x1 , x2 , . . . , xn ) be a smooth coordinate system for M , defined
over some open subset U of M , let gij be the components of the metric
tensor g with respect to this coordinate system, so that
∂
∂
gij = g
,
,
∂xi ∂xj
and let g ij be the smooth functions on U defined such that
n
P
gij g jk = δik ,
j=1
where δik denotes the Kronecker delta.
Let Xp be a tangent vector at some point p of U , and let
n
X
∂ j
Xp =
a
.
j
∂x
p
j=1
When using traditional index notation it is customary to denote the components of Xp[ by a1 , a2 , . . . , an , so that
Xp[ =
n
X
ak dxkp .
k=1
Similarly given an element θp of the cotangent space Tp∗ M at p, where
θp =
n
X
ck dxkp ,
k=1
it is customary to denote the components of θp] by c1 , c2 , . . . , xn , so that
n
X
∂ ]
j
.
θp =
c
j
∂x
p
j=1
It follows from Lemma 9.2 that
ak =
n
X
gjk aj
and cj =
j=1
n
X
k=1
177
g jk ck .
It is also common practice, when using traditional index notation for
tensor fields, to adopt analogous operations of raising and lowering indices
in order to convert between tensors of different types.
Example Let S be a tensor field S of type (1, 3) on a Riemannian or pseudoRiemannian manifold M . Then S is represented at each point p of M by a
trilinear map S: Tp M × Tp M × Tp M → Tp M . Now the metric tensor g on M
determines an isomorphism between the vector space of such trilinear maps
and the vector space of quadrilinear forms on Tp M . This isomorphism sends
the trilinear map S to the quadrilinear form S [ , where
S [ (Wp , Xp , Yp , Zp ) = g(Wp , R(Xp , Yp , Zp ))
for all Wp , Xp , Yp , Zp ∈ Tp M . If
S=
n
X
S l ijk dxi ⊗ dxj ⊗ dxk ⊗
l,i,j,k=1
then
[
S =
n
X
∂
∂xl
Shijk dxh ⊗ dxi ⊗ dxj ⊗ dxk ,
h,i,j,k=1
where
Shijk =
n
X
ghl S l ijk .
l=1
Then
S l ijk =
n
X
g lh Shijk ,
h=1
where the functions g mh are defined such that
n
P
g mh ghl = δlm .
h=1
9.2
The Levi-Civita Connection
Let M be a Riemannian or pseudo-Riemannian manifold, with metric tensor g, and let ∇ be an affine connection on M . We say that ∇ is compatible
with the metric tensor g if
Z[g(X, Y )] = g(∇Z X, Y ) + g(X, ∇Z Y )
for all smooth vector fields X, Y and Z on M . We shall show that on every
pseudo-Riemannian manifold there exists a unique torsion-free connection
that is compatible with the metric tensor.
178
Lemma 9.3 Let M be a Riemannian or pseudo-Riemannian manifold with
metric tensor g, and let ∇ be a torsion-free affine connection on M that is
compatible with the metric tensor g. Then
2g(∇X Y, Z) = X[g(Y, Z)] + Y [g(X, Z)] − Z[g(X, Y )]
+ g([X, Y ], Z) − g([X, Z], Y ) − g([Y, Z], X)
Proof Let X, Y and Z be smooth vector fields on M . The requirement that
∇ be both torsion-free and compatible with the metric tensor ensures that
g([X, Y ], Z)
g([Y, Z], X)
g([Z, X], Y )
X[g(Y, Z)]
Y [g(Z, X)]
Z[g(X, Y )]
=
=
=
=
=
=
g(∇X Y, Z) − g(∇Y X, Z),
g(∇Y Z, X) − g(∇Z Y, X),
g(∇Z X, Y ) − g(∇X Z, Y ),
g(∇X Y, Z) + g(∇X Z, Y ),
g(∇Y Z, X) + g(∇Y X, Z),
g(∇Z X, Y ) + g(∇Z Y, X).
Thus if
χ(X, Y, Z) =
1
2
X[g(Y, Z)] + Y [g(X, Z)] − Z[g(X, Y )]
+ g([X, Y ], Z) − g([X, Z], Y ) − g([Y, Z], X)
then
2χ(X, Y, Z) = g(∇X Y, Z) + g(∇X Z, Y ) + g(∇Y Z, X) + g(∇Y X, Z)
− g(∇Z X, Y ) − g(∇Z Y, X) + g(∇X Y, Z) − g(∇Y X, Z)
+ g(∇Z X, Y ) − g(∇X Z, Y ) − g(∇Y Z, X) + g(∇Z Y, X)
= 2g(∇X Y, Z).
The result follows directly.
Lemma 9.3 shows that a Riemannian or pseudo-Riemannian manifold can
have at most one torsion-free affine connection that is compatible with the
metric tensor. We now show that there exists a smooth affine connection on
any Riemannian or pseudo-Riemannian manifold which is characterized by
the identity given in the statement of Lemma 9.3.
179
Theorem 9.4 Let (M, g) be a Riemannian or pseudo-Riemannian manifold.
Then there exists a unique torsion-free affine connection ∇ on M compatible
with the metric tensor g. This connection is characterized by the identity
2g(∇X Y, Z) = X[g(Y, Z)] + Y [g(X, Z)] − Z[g(X, Y )]
+ g([X, Y ], Z) − g([X, Z], Y ) − g([Y, Z], X)
for all smooth vector fields X, Y and Z on M .
Proof Given smooth vector fields X, Y and Z on M , let χ(X, Y, Z) be the
smooth function on M defined by
χ(X, Y, Z) =
1
(X[g(Y, Z)]
2
+ Y [g(X, Z)] − Z[g(X, Y )]
+ g([X, Y ], Z) − g([X, Z], Y ) − g([Y, Z], X)).
Then
χ(X1 + X2 , Y, Z) = χ(X1 , Y, Z) + χ(X2 , Y, Z),
χ(X, Y1 + Y2 , Z) = χ(X, Y1 , Z) + χ(X, Y2 , Z),
χ(X, Y, Z1 + Z2 ) = χ(X, Y, Z1 ) + χ(X, Y, Z2 )
for all smooth vector fields X, X1 , X2 , Y , Y1 , Y2 , Z, Z1 and Z2 on M . Now
[X, f Y ] = f [X, Y ] + X[f ] Y
and [f X, Y ] = f [X, Y ] − Y [f ] X
for all smooth real-valued functions f and smooth vector fields X and Y on
M (see Lemma 7.6). It follows that
1
χ(f X, Y, Z) = f χ(X, Y, Z) + 2 Y [f ] g(X, Z) − Z[f ] g(X, Y )
− Y [f ] g(X, Z) + Z[f ] g(Y, X)
= f χ(X, Y, Z),
1
2
X[f ] g(Y, Z) − Z[f ] g(X, Y )
+ X[f ] g(Y, Z) + Z[f ] g(Y, X)
χ(X, f Y, Z) = f χ(X, Y, Z) +
= f χ(X, Y, Z) + X[f ] g(Y, Z),
χ(X, Y, f Z) = f χ(X, Y, Z) + 21 X[f ] g(Y, Z) + Y [f ] g(X, Z)
− X[f ] g(Z, Y ) − Y [f ] g(Z, X)
= f χ(X, Y, Z)
180
for all smooth real-valued functions f and smooth vector fields X, Y and
Z on M . An application of Corollary 6.16 shows that each smooth vector
field Y on M determines a well-defined bilinear form µY : Tp M ×Tp M → R on
the tangent space Tp M at each point p of M characterized by the property
that χ(X, Y, Z) = µY (Xp , Zp ) for all smooth vector fields X and Z on M .
Given a smooth vector field Y defined around a point p of M , and given a
tangent vector Xp ∈ Tp M at p, let µXp ,Y be the element of the cotangent
space Tp∗ M defined such that hµXp ,Y , Zp i = µY (Xp , Zp ) for all Zp ∈ Tp M ,
and let ∇X p Y = µ]Xp ,Y , so that
g(∇X p Y, Zp ) = g(µ]Xp ,Y , Zp ) = hµXp ,Y , Zp i = µY (Xp , Zp )
for all Zp ∈ Tp M . Clearly
∇Wp +Xp Y = ∇Wp Y + ∇Xp Y
and ∇c Xp Y = c ∇Xp Y
for all Wp , Xp ∈ Tp M and for all real numbers c. The identity
χ(X, f Y, Z) = f χ(X, Y, Z) + X[f ] g(Y, Z)
ensures that
∇Xp (Y + Z) = ∇Xp Y + ∇Xp Z
and ∇Xp (f Y ) = Xp [f ] Y + f ∇Xp Y
for all smooth real-valued functions f and smooth vector fields Y and Z
defined around the point p. Moreover g(∇X Y, Z) = χ(X, Y, Z), and therefore
g(∇X Y, Z) is a smooth real-valued function, and therefore ∇X Y is a smooth
vector field, for all smooth vector fields X, Y and Z defined around the
point p. We have thus shown that the differential operator ∇ is a smooth
affine connection on M .
Let X, Y and Z be smooth vector fields on M . Then
χ(X, Y, Z) − χ(Y, X, Z) = g([X, Y ], Z).
It follows that ∇X Y − ∇Y X = [X, Y ]. This shows that the affine connection
∇ is torsion-free. Also
g(∇X Y, Z) + g(Y, ∇X Z) = χ(X, Y, Z) + χ(X, Z, Y ) = X[g(Y, Z)],
and thus the affine connection ∇ is compatible with the metric tensor.
Lemma 9.3 guarantees that this torsion-free affine connection compatible
with the metric tensor is uniquely determined, as required.
181
Definition Let M be a Riemannian or pseudo-Riemannian manifold. The
Levi-Civita connection on M is the unique smooth torsion-free affine connection on M that is compatible with the metric tensor on M .
Example Let M be a smooth n-dimensional submanifold of k-dimensional
Euclidean space Rk . Given (tangential) vector fields X and Y on M , we
decompose the directional derivative ∂X Y of Y along X as ∂X Y = ∇X Y −
S(X, Y ), where ∇X Y is tangential to M and S(X, Y ) is orthogonal to M .
Then ∇ is a torsion-free affine connection on M . Now the restriction to the
tangent spaces of M of the standard scalar product h., .i on Rk defines a
Riemannian metric g on M . Moreover
g(∇X Y, Z) + g(Y, ∇X Z) = h∂X Y, Zi + hY, ∂X Zi = X[hY, Zi] = X[g(Y, Z)]
for all vector fields X, Y and Z on M that are everywhere tangential to M .
We conclude that the affine connection ∇ is the Levi-Civita connection of
the Riemannian manifold M .
Corollary 9.5 Let M be a Riemannian or pseudo-Riemannian manifold
with metric tensor g, and let (x1 , x2 , . . . , xn ) be a smooth coordinate system
for M defined over an open subset U of M , let
∂
∂
,
gij = g
∂xi ∂xj
for i, j = 1, 2, . . . , n, and let g kl be the smooth functions defined on U so that
n
P
g kl = g lk and
gij g jk = δik , where δik denotes the Kronecker delta. Let ∇ be
j=1
the Levi-Civita connection on M . Then
n
X
∂
∂
∇ ∂j k =
Γijk i
∂x ∂x
∂x
i=1
where
Γijk
1
= g im
2
∂gmk ∂gjm ∂gjk
+
− m
∂xj
∂xk
∂x
.
Moreover Γijk = Γikj for all i, j and k, and if X and Y are smooth vector
fields on U , and if
X=
n
X
i=1
then
∇X Y =
∂
v
,
∂xi
n
X
i,j=1
i
Y =
n
X
i=1
n
wi
∂
,
∂xi
∂wi X i j k
vj j +
Γjk v w
∂x
k=1
182
!
∂
.
∂xi
Proof The basis vector fields determined by the coordinate system satisfy
∂
∂
=0
,
∂xj ∂xk
for all j and k. It follows from Theorem 9.4 that
∂
1 ∂gmk ∂gjm ∂gjk
∂
g ∇ ∂j k ,
=
+
− m .
∂x ∂x
∂xm
2 ∂xj
∂xk
∂x
A straightforward application of Lemma 9.2 yields the formula for the quantities Γijk . Then
∂
∂
∂
∂
i
i
i
i
Γjk − Γkj = dx , ∇ ∂ j k − ∇ ∂
= dx ,
,
= 0.
∂x ∂x
∂xk ∂xj
∂xj ∂xk
The formula giving an expression for ∇X Y in terms the components of the
vector fields X and Y in a local coordinates system is a particular case of
the more general result proved in Lemma 8.10.
The quantities Γijk that represent the Levi-Civita connection of a Riemannian or pseudo-Riemannian manifold with respect to a smooth local
coordinate system on that manifold are known as Christoffel symbols, and
can be calculated from the components of the metric tensor and their first
derivatives according to the formula given in Corollary 9.5
9.3
The Riemann Curvature Tensor
Definition Let M be a Riemannian or pseudo-Riemannian manifold with
metric tensor g. The Riemann curvature tensor R of M is defined by the
formula
R(W, Z, X, Y ) = g(W, R(X, Y )Z),
where
R(X, Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z
for all smooth vector fields X, Y and Z on M , where ∇ denotes the LeviCivita connection on M .
The Riemann curvature tensor on a Riemannian or pseudo-Riemannian
manifold M is thus the curvature tensor of the Levi-Civita connection on M .
The value of R(W, Z, X, Y ) at a point p of M is determined by the values of
the vector fields W , X, Y and Z at that point.
The following proposition is a special case of Proposition 8.11.
183
Proposition 9.6 . Let M be a Riemannian or pseudo-Riemannian manifold
with metric tensor g, let (x1 , x2 , . . . , xn ) be a smooth coordinate system for
M defined over an open subset U of M , and let Γijk denote the Christoffel
symbols that represent the Levi-Civita connection ∇ determined by the metric
tensor g with respect to the smooth local coordinates x1 , x2 , . . . , xn , so that
n
∂ X
∂
∇ ∂
=
Γljk l ,
k
∂x
∂x
l=1
∂xj
on U . Let Rl ijk denote the components of the Riemann curvature tensor with
respect to this coordinate system, so that
R=
n
X
Rl ijk
l=1
∂
⊗ dxi ⊗ dxj ⊗ dxk
∂xl
on U . Then
Rl ijk =
n
∂Γlki ∂Γlji X l m
−
+
(Γ Γ − Γlkm Γm
ji )
∂xj
∂xk m=1 jm ki
for l, i, j, k = 1, 2, . . . , n. Moreover if X, Y and Z are smooth vector fields
over U , and if
X=
n
X
i=1
∂
,
u
∂xi
i
n
X
∂
Y =
v
,
∂xj
j=1
then
R(Y, Z)X =
n
X
j
Z=
Rl ijk ui v j wk
i,j,k,l=1
n
X
k=1
wk
∂
,
∂xk
∂
.
∂xl
Proposition 9.7 Let M be a Riemannian or pseudo-Riemannian manifold.
Then the Riemann curvature tensor R on M satisfies the following identities
at each point p of M , and for all W, X, Y, Z ∈ Tp M :—
(i) R(W, Z, X, Y ) = −R(W, Z, Y, X);
(ii) R(W, X, Y, Z) + R(W, Y, Z, X) + R(W, Z, X, Y ) = 0;
(iii) R(W, Z, X, Y ) = −R(Z, W, X, Y );
(iv) R(W, Z, X, Y ) = R(X, Y, W, Z).
184
Proof Property (i) follows directly from the definition of the Riemann curvature tensor, and (ii) corresponds to the First Bianchi Identity
R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0
(see Proposition 8.14). Now
X[Y [g(W, Z)]] = X [g(∇Y W, Z) + g(W, ∇Y Z)]
= g(∇X ∇Y W, Z) + g(∇Y W, ∇X Z)
+ g(∇X W, ∇Y Z) + g(W, ∇X ∇Y Z),
and hence
[X, Y ][g(W, Z)] = X[Y [g(W, Z)]] − Y [X[g(W, Z)]]
= g(∇X ∇Y W − ∇Y ∇X W, Z)
+ g(W, ∇X ∇Y Z − ∇Y ∇X Z).
Therefore
R(W, Z, X, Y ) + R(Z, W, X, Y )
= g(W, R(X, Y )Z) + g(R(X, Y )W, Z)
= [X, Y ][g(W, Z)] − g(∇[X,Y ] W, Z) − g(W, ∇[X,Y ] Z)
= 0.
This proves (iii). Using (i), (ii) and (iii), we see that
2R(W, Z, X, Y ) = R(W, Z, X, Y ) − R(Z, W, X, Y )
= −R(W, X, Y, Z) − R(W, Y, Z, X)
+ R(Z, X, Y, W ) + R(Z, Y, W, X)
= (R(X, W, Y, Z) + R(X, Z, W, Y ))
+ (R(Y, W, Z, X) + R(Y, Z, X, W ))
= −R(X, Y, Z, W ) − R(Y, X, W, Z)
= 2R(X, Y, W, Z).
This proves (iv).
The following result expressed the properties of the Riemann curvature
tensor in terms of its components with respect to a smooth local coordinate system on the Riemannian or pseudo-Riemannian manifold. It follows
directly from Proposition 9.7
185
Corollary 9.8 Let M be a Riemannian or pseudo-Riemannian manifold
with metric tensor g, and let the functions Rhijk be the components of the
Riemann curvature tensor R of M , determined with respect to a smooth local
coordinate system (x1 , x2 , . . . , xn ) for M defined over an open subset U of M ,
so that
∂
∂
∂
∂
Rhijk = g
.
,R
,
,
∂xh
∂xj ∂xk
∂xi
Then these components have the following properties:—
(i) Rhijk = −Rhikj ;
(ii) Rhijk + Rhjki + Rhkij = 0;
(iii) Rhijk = −Rihjk ;
(iv) Rhijk = Rjkhi .
The curvature tensor of a Riemannian or pseudo-Riemannian manifold
also satisfies the Second Bianchi Identity
(∇X R)(Y, Z)W + (∇Y R)(Z, X)W + (∇Z R)(X, Y )W = 0
(see Corollary 8.16).
9.4
The Sectional Curvatures of a Riemannian Manifold
Let M be a Riemannian manifold, let p be a point of M , and let P be a
two-dimensional vector subspace of the tangent space Tp M to M at p. Let
(E1 , E2 ) be an orthonormal basis of P . We define the sectional curvature
K(P ) of M in the plane P by the formula
K(P ) = R(E1 , E2 , E1 , E2 ).
Note that if X and Y are tangent vectors contained in the plane P then
X = a11 E1 + a12 E2 ,
Y = a21 E1 + a22 E2 ,
for some real numbers a11 , a12 , a21 and a22 , and hence
R(X, Y, X, Y ) =
=
=
=
R(X, Y, a11 E1 + a12 E2 , a21 E1 + a22 E2 )
(a11 a22 − a12 a21 )R(X, Y, E1 , E2 )
(det A)R(X, Y, E1 , E2 ) = (det A)2 R(E1 , E2 , E1 , E2 )
(det A)2 K(P ),
186
where A is the matrix given by
A=
a11
a21
a12
a22
.
In particular, if (X, Y ) is any orthonormal basis of P then the matrix A is
an orthogonal matrix, and thus det A = ±1. It follows that the value of the
sectional curvature K(P ) does not depend on the choice of the orthonormal
basis (E1 , E2 ) of P .
Lemma 9.9 Let M be a Riemannian manifold with metric tensor g, and
let p be a point of M . Then the values of the sectional curvatures K(P ) for
all two-dimensional vector subspaces P of the tangent space Tp M to M at p
determine the Riemann curvature tensor at p.
Proof The calculation given above shows that the sectional curvatures determine the values of R(X, Y, X, Y ) for all X, Y ∈ Tp M .
Now suppose that we are given X, Y, Z ∈ Tp M . Using the symmetries of
the Riemann curvature tensor listed in Proposition 9.7, we see that
2R(X, Y, X, Z) = R(X, Y, X, Z) + R(X, Z, X, Y )
= R(X, Y + Z, X, Y + Z) − R(X, Y, X, Y )
− R(X, Z, X, Z).
Thus the sectional curvatures K(P ) determine the values of R(X, Y, X, Z)
for all tangent vectors X, Y and Z at p. It follows from this that the sectional
curvatures determine R(X, Y, Z, X), R(Y, X, X, Z) and R(Y, X, Z, X). But
3R(W, X, Y, Z) = 2R(W, X, Y, Z) − R(W, Y, Z, X) − R(W, Z, X, Y )
= (R(W, X, Y, Z) + R(W, Y, X, Z))
+ (R(W, X, Y, Z) + R(W, Z, Y, X))
= R(W, X + Y, X + Y, Z) − R(W, X, X, Z)
− R(W, Y, Y, Z) + R(W, X + Z, Y, X + Z)
− R(W, X, Y, X) − R(W, Z, Y, Z).
We conclude that R(W, X, Y, Z) is determined by the sectional curvatures of
M , as required.
187
10
10.1
Covariant Derivatives along Curves and
Surfaces
Vector Fields along Smooth Maps
Definition Let Q and M be smooth manifolds, and let ϕ: Q → M be a
smooth map. A vector field V along the map ϕ is a function V : Q → T M
from Q to the total space T M of the tangent bundle πT M : T M → M of M
with the property that πT M ◦ V = ϕ.
Let Q and M be smooth manifolds. A smooth vector field V : Q → T M
along a smooth map ϕ: Q → M is thus a smooth map from Q to T M which
associates to each point q of Q a tangent vector V (q) to the manifold M at
the point ϕ(q).
Let (x1 , x2 , . . . , xn ) be a smooth coordinate system defined over some open
set U in M . Given any smooth vector field V : Q → T M along the smooth
map ϕ: Q → M , there exist smooth real-valued functions v 1 , v 2 , . . . , v n on Q
such that
n
X
∂ i
V (q) =
v (q) i ∂x ϕ(q)
i=1
for all q ∈ ϕ−1 (U ).
We say that a vector field V along the map ϕ is smooth if, given any
smooth coordinate system (x1 , x2 , . . . , xn ) defined over an open subset U of
M , the components v 1 , v 2 , . . . , v n of V with respect to this coordinate system
are smooth functions on ϕ−1 (U ). In particular, one can define in this way
smooth vector fields along curves and surfaces in the smooth manifold M .
Example Let γ: I → M be a smooth curve in the smooth manifold M ,
defined on some open interval I in R. Then a vector field V along the
curve γ is a function which associates to each t ∈ I a tangent vector V (t) to
M at γ(t). The map that sends t ∈ I to the velocity vector γ 0 (t) of the curve
at time t is a smooth vector field along γ.
10.2
Moving Frames
Definition Let M be a smooth manifold of dimension n, and let U be an
open set in M . A moving frame over U is an n-tuple of smooth vector fields
E1 , E2 , . . . , En over U such that, for each point p of U , the values
(E1 )p , (E2 )p , . . . , (En )p
188
of these vector fields at the point p constitute a basis for the tangent Tp M
at that point.
Example Let M be a smooth manifold, and let U be the domain of a smooth
coordinate system (x1 , x2 , . . . , xn ) for M . Then the vector fields
∂
∂
∂
,
,...
1
2
∂x
∂x
∂xn
constitute a moving frame over the open set U .
10.3
Covariant Differentiation of Vector Fields along
Curves
Let M be a smooth manifold, and let γ: I → M be a smooth curve in the
smooth manifold M , defined on some open interval I in R. Let t0 ∈ I, and
let E1 , E2 , . . . , En be a moving frame for M defined around the point γ(t0 ) of
M . Given any smooth vector field V along the curve γ, there exist smooth
real-valued functions v 1 , v 2 , . . . , v n on γ −1 (U ) such that
V (t) =
n
X
v j (t)(Ej )γ(t) ,
j=1
for all t ∈ γ −1 (U ).
Let ∇ be an affine connection on M . Given any smooth vector field V
DV (t)
along the smooth curve γ we wish to define the covariant derivative
dt
of the vector field V along the curve. Moreover this covariant derivative
operator acting on vector fields along smooth curves should be determined
in some natural fashion by the affine connection ∇.
Let us first consider the particular case where V (t) = Yγ(t) for all t ∈ I,
where Y is some smooth vector field on M . Then there exist smooth realvalued functions w1 , w2 , . . . , wn on U such that
Y (p) =
n
X
wj (p)(Ej )p
j=1
for all p ∈ U . Then
V (t) =
n
X
v j (t)(Ej )γ(t) ,
j=1
189
for all t ∈ γ −1 (U ), where hj (t) = wj (γ(t)) for j = 1, 2, . . . , n. Now
n X
dwj (γ(t))
j
∇γ 0 (t) Y =
(Ej )γ(t) + w (γ(t))∇γ 0 (t) Ej
dt
j=1
n X
dhj (t)
j
=
(Ej )γ(t) + h (t)∇γ 0 (t) Ej
dt
j=1
for all t ∈ γ −1 (U ).
This suggests that it might be reasonable to define the covariant derivative
dV (t)
of any smooth vector field V along the curve γ so that if
dt
V (t) =
n
X
v j (t)(Ej )γ(t) ,
j=1
where h1 , h2 , . . . , hn are smooth real-valued functions on γ −1 (U ), then
n DV (t) X dhj (t)
j
=
(Ej )γ(t) + h (t)∇γ 0 (t) Ej
dt
dt
j=1
for all t ∈ γ −1 (U ). However we need to verify that the value of the right hand
side of the above equation is completely determined by the smooth vector
field V along γ and the affine connection ∇ on M , and does not does not
depend on the choice of the moving frame E1 , E2 , . . . , En .
Let Ê1 , Ê2 , . . . , Ên be another smooth moving frame for M defined over
an open subset Û of M , where U ∩ Û ∩ γ(I) is non-empty. Then there exist
smooth real-valued functions Aj k on U ∩ Û such that
Êk =
n
X
Aj k Ej
j=1
on U ∩ Û . For all p ∈ U ∩ Û , let A(p) be the n × n matrix whose entry
in the jth row and kth column is Aj k (p). Then this matrix A(p) is a nonsingular matrix. Now, given any smooth vector field V along the smooth
curve γ: I → M , there exist smooth real-valued functions h1 , h2 , . . . , hn on
γ −1 (U ) and smooth real-valued functions ĥ1 , ĥ2 , . . . , ĥn on γ −1 (Û ) such that
n
n
P
P
V (t) =
hj (t)(Ej )γ(t) on γ −1 (U ) and V (t) =
ĥk (t)(Êk )γ(t) on γ −1 (Û ).
j=1
j=1
Then
j
h (t) =
n
X
Aj k (γ(t))ĥk (t)
k=1
190
for all q ∈ U ∩ Û . Then
n
dhj (t) X
=
dt
k=1
d(Aj k (γ(t)) k
dĥk (t)
ĥ (t) + Aj k (γ(t))
dt
dt
!
.
It follows that
n X
dhj (t)
j
(Ej )γ(t) + h (t)∇γ 0 (t) Ej
dt
j=1
=
n X
d(Aj k (γ(t))
dt
j,k=1
j
ĥk (t)(Ej )γ(t) + Aj k (γ(t))
k
+ A k (γ(t))ĥ (t)∇γ 0 (t) Ej
=
n
X
dĥk (t)
dt
k=1
+
n
X
j,k=1
=
k=1
=
n
X
k=1
dt
(Êk )γ(t)
ĥk (t)
n
X
dĥk (t)
dĥk (t)
(Ej )γ(t)
dt
d(Aj (γ(t))
k
(Ej )γ(t) + Aj k (γ(t))∇γ 0 (t) Ej
dt
(Êk )γ(t) +
n
X
ĥk (t)∇γ 0 (t) (Aj k (γ(t))Ej )
j,k=1
n
X
dĥk (t)
ĥk (t)∇γ 0 (t) Êk
(Êk )γ(t) +
dt
j,k=1
!
.
We may therefore employ the formula
n X
dhj (t)
j=1
dt
j
(Ej )γ(t) + h (t)∇γ 0 (t) Ej
on order to define the covariant derivative of the vector field on γ −1 (U ),
since the tangent vector to M at γ(t) determined by this expression does not
depend on the choice of moving frame used when evaluating this expression.
Definition Let M be a smooth manifold, let ∇ be a smooth affine connection on M , let γ: I → M be a smooth curve in M defined on some open
interval I in R, and let V : I → T M be a smooth vector field along the
DV (t)
curve γ. The covariant derivative
of the vector field V is the vector
dt
field determined, for values of t sufficiently close to some given value t0 , by
191
the equation
n
DV (t) X
=
dt
j=1
dhj (t)
j
(Ej )γ(t) + h (t)∇γ 0 (t) Ej ,
dt
where E1 , E2 , . . . , En is a moving frame for the smooth manifold M defined
on some open neighbourhood U of the point γ(t0 ) and h1 , h2 , . . . , hn are the
smooth functions on γ −1 (U ) determined such that
V (t) =
n
X
v j (t)(Ej )γ(t) ,
j=1
for all t ∈ γ −1 (U ).
Lemma 10.1 Let M be a smooth manifold, let ∇ be an affine connection on
M , and let γ: I → M be a smooth curve in M . Let V and W be smooth vector
fields along γ and let f : I → R be a smooth real-valued function. Then
(i)
DV (t) DW (t)
D(V (t) + W (t))
=
+
,
dt
dt
dt
(ii)
D(f (t)V (t))
df (t)
DV (t)
=
V (t) + f (t)
,
dt
dt
dt
(iii) if V (t) = Xγ(t) for all t, where X is some smooth vector field defined
DV (t)
over an open set in M , then
= ∇γ 0 (t) X.
dt
Moreover the differential operator D/dt is the unique operator on the space
of smooth vector fields along the curve γ satisfying (i), (ii) and (iii).
Definition A smooth vector field V along a smooth curve γ is said to be
DV (t)
parallel along γ if
= 0 for all t.
dt
10.4
Vector Fields along Parameterized Surfaces
Let M be a smooth manifold, let U be a connected open set in Rm , and let
ϕ: U → M be a smooth map from U to M . Given (t1 , t2 , . . . , tm ) ∈ U , we
define
∂ϕ(t1 , t2 , . . . , tm )
∂ti
to be the velocity vector of the curve t 7→ ϕ(t1 , . . . , ti−1 , t, ti+1 , . . . , tm ) at t =
ti . Then ∂ϕ/∂ti is a smooth vector field along the map ϕ for i = 1, 2, . . . , m.
192
Let ∇ be an affine connection on M . Given any smooth vector field V
along the map ϕ, and given (t1 , t2 , . . . , tm ) ∈ U , we define
DV (t1 , t2 , . . . , tm )
∂ti
to be the covariant derivative of the vector field
t 7→ V (t1 , . . . , ti−1 , t, ti+1 , . . . , tm )
along the curve t 7→ ϕ(t1 , . . . , ti−1 , t, ti+1 , . . . , tm ) at t = ti . Then the partial
covariant derivative DV /∂ti is a smooth vector field along the map ϕ of
i = 1, 2, . . . , m.
Let M be a smooth manifold of dimension n. A smooth parameterized
surface in M is a smooth map ϕ: U → M defined on a connected open
subset U on R2 .
Lemma 10.2 Let M be a smooth manifold and let ∇ be an affine connection
on M . Let V be a smooth vector field along a smooth parameterized surface
ϕ: U → M in M . Then
∂ϕ(s, t) ∂ϕ(s, t) D ∂ϕ(s, t) D ∂ϕ(s, t)
−
= T
,
,
∂s ∂t
∂t ∂s
∂s
∂t
∂ϕ ∂ϕ
D DV (s, t) D DV (s, t)
−
= R
,
V (s, t),
∂s
∂t
∂t
∂s
∂s ∂t
where T and R are the torsion and curvature tensors of the affine connection ∇.
Proof Without loss of generality, we may suppose that the image of the map
ϕ: U → M is contained in the domain of some smooth coordinate system
(x1 , x2 , . . . , xn ). Let B1 , B2 , . . . , Bn be the smooth vector fields over this
coordinate patch defined by
∂
Bi =
∂xi
for i = 1, 2, . . . , n. Then the vector fields B1 , B2 , . . . , Bn constitute a moving
frame defined over some open set in M that contains ϕ(U ). Moreover
[Bj , Bk ] = 0
for j, k = 1, 2, . . . , n, and therefore
∇Bj Bk − ∇Bk Bj = T (Bj , Bk ),
∇Bj ∇Bk Bi − ∇Bk ∇Bj Bi = R(Bj , Bk )Bi ,
193
for i, j, k = 1, 2, . . . , n.
The map ϕ: U → M is specified, with respect to the coordinate system
1
(x , x2 , . . . , xn ), by smooth real-valued functions ϕ1 , ϕ2 , . . . , ϕn on U , where
ϕi (s, t) = xi (ϕ(s, t) for i = 1, 2, . . . , n and for all s, t ∈ U . It follows that
n
n
∂ϕ X ∂ϕj
=
Bj ,
∂s
∂s
j=1
Thus
∂ϕ X ∂ϕk
=
Bk .
∂t
∂t
k=1
n
n
DX X ∂ϕj
=
∇Bj X,
∂s
∂s
j=1
DX X ∂ϕk
=
∇Bk X.
∂t
∂t
k=1
for all smooth vector fields X on M defined around points of ϕ(U ). Now
n
X
D ∂ϕk D ∂ϕ
=
Bk
∂s ∂t
∂s
∂t
k=1
=
n
X
∂ 2 ϕk
k=1
n
X
∂ϕj ∂ϕk
Bk +
∇Bj Bk .
∂s∂t
∂s
∂t
j,k=1
Thus
n
X
D ∂ϕ D ∂ϕ
∂ϕj ∂ϕk
−
=
∇Bj Bk − ∇Bk Bj
∂s ∂t
∂t ∂s
∂s ∂t
j,k=1
=
n
∂ϕ ∂ϕ X
∂ϕj ∂ϕk
T (Bj , Bk ) = T
,
.
∂s
∂t
∂s
∂t
j,k=1
Let f : U → R be a smooth real-valued function on U , and let V be a
smooth vector field along the map ϕ. Then
D D(f V )
D ∂f
DV =
V +f
∂s ∂t
∂s ∂t
∂t
2
∂ f
∂f DV
∂f DV
D DV
=
V +
+
+f
,
∂s∂t
∂t ∂s
∂s ∂t
∂s ∂t
and thus
D D
D DV
D D
D DV −
(f V ) = f
−
∂s ∂t ∂t ∂s
∂s ∂t
∂t ∂s
Now any smooth vector field V along the map ϕ can be expressed in the
form
n
X
V (s, t) =
v i (s, t)(Bi )ϕ(s,t)
i=1
194
for some smooth real-valued functions v 1 , v 2 , . . . , v n on U . It follows that
n
X D D
D DV
D DV
D D
−
=
vi
−
Bi .
∂s ∂t
∂t ∂s
∂s
∂t
∂t
∂s
i=1
But
n
X
DD
D ∂ϕk
Bi =
∇Bk Bi
∂s ∂t
∂s ∂t
k=1
=
n
X
∂ 2 ϕk
k=1
∂s∂t
∇Bk Bi +
n
X
∂ϕj ∂ϕk
∇Bj ∇Bk Bi ,
∂s
∂t
j,k=1
and hence
n
D D
X
D D
∂ϕj ∂ϕk
−
Bi =
∇Bj ∇Bk Bi − ∇Bk ∇Bj Bi
∂s ∂t ∂t ∂s
∂s ∂t
j,k=1
n
X
∂ϕj ∂ϕk
R(Bj , Bk )Bi
=
∂s
∂t
j,k=1
∂ϕ ∂ϕ = R
,
Bi .
∂s ∂t
We deduce that
n
∂ϕ ∂ϕ ∂ϕ ∂ϕ X
D DV
D DV
−
=
viR
,
Bi = R
,
V,
∂s ∂t
∂t ∂s
∂s ∂t
∂s ∂t
i=1
as required.
195
11
11.1
Geodesics and Jacobi Fields
Geodesics
Definition Let M be a Riemannian or pseudo-Riemannian manifold which
is provided with a smooth affine connection ∇, and let γ: I → M be a smooth
curve in M , defined over some interval I in R. We say that γ is a geodesic
(with respect to the connection ∇ if and only if
D dγ(t)
= 0.
dt
dt
Thus γ is a geodesic if and only if the velocity vector field t 7→ γ 0 (t) is parallel
along γ (with respect to the connection ∇ on M ). The geodesic γ: I → M is
said to be maximal if it cannot be extended to a geodesic defined over some
interval J, where I ⊂ J and I 6= J.
A smooth curve in a Riemannian or pseudo-Riemannian manifold is said
to be a geodesic if it is a geodesic with respect to the Levi-Civita connection
determined by the metric tensor on the manifold.
Covariant differentiation of smooth vector fields along curves and surfaces
in a Riemannian or pseudo-Riemannian manifold is defined with respect to
the Levi-Civita connection determined by the metric tensor on the manifold.
Lemma 11.1 Let M be a Riemannian or pseudo-Riemannian manifold, and
let g denote the metric tensor on M . Let γ: I → M be a smooth curve in M ,
defined on an open interval I in R, and let V : I → T M and W : I → T M be
smooth vector fields along the curve γ. Then
DV (t)
DW (t)
d
g(V (t), W (t)) = g
, W (t) + g V (t),
.
dt
dt
dt
Proof Let E1 , E2 , . . . , En be a moving frame on M defined over an open
neighbourhood U of γ(t0 ) for some t0 ∈ I. Then there are smooth functions
v 1 , v 2 , . . . , v n and w1 , w2 , . . . , wn on γ −1 (U ) defined such that
V (t) =
n
X
j
v (Ej )γ(t) ,
W (t) =
j=1
n
X
wk (Ek )γ(t)
k=1
for all t ∈ γ −1 (U ). Then
n
DV (t) X
=
dt
j=1
dv j (t)
(Ej )γ(t) + v j (t) ∇γ 0 (t) Ej
dt
196
and
n
DW (t) X
=
dt
k=1
dwk (t)
k
(Ek )γ(t) + w (t) ∇γ 0 (t) Ek .
dt
Then
DV (t)
DW (t)
g
, W (t) + g V (t),
dt
dt
n
X dv j (t)
=
wk (t) g((Ej )γ(t) , (Ek )γ(t) )
dt
j,k=1
+ v j (t) wk (t) g(∇γ 0 (t) Ej , (Ek )γ(t) )
dwk (t)
+ v j (t)
g((Ej )γ(t) , (Ek )γ(t) )
dt
+ v j (t) wk (t) g((Ej )γ(t) , ∇γ 0 (t) Ek )
n X
d =
j,k=1
dt
v (t)w (t) g((Ej )γ(t) , (Ek )γ(t) )
j
k
+ v j (t) wk (t) g(∇γ 0 (t) Ej , (Ek )γ(t) ) + g((Ej )γ(t) , ∇γ 0 (t) Ek )
n X
d =
j,k=1
dt
v j (t)wk (t) g((Ej )γ(t) , (Ek )γ(t) )
d g((Ej )γ(t) , (Ek )γ(t) )
+ v (t) w (t)
dt
n
X
d j
=
v (t)wk (t) g((Ej )γ(t) , (Ek )γ(t) )
dt
j,k=1
d
=
g(V (t), W (t)) ,
dt
j
k
as required.
Lemma 11.2 Let M be a Riemannian or pseudo-Riemannian manifold, and
let g denote the metric tensor on M . Let γ: I → M be a geodesic in M . Then
d
g(γ 0 (t), γ 0 (t)) = 0,
dt
and thus g(γ 0 (t), γ 0 (t)) is constant along the geodesic.
Proof
d
d
g(γ 0 (t), γ 0 (t)) =
g
dt
dt
dγ(t) dγ(t)
,
dt
dt
197
= g
D dγ dγ
,
dt dt dt
+g
dγ D dγ
,
dt dt dt
= 0,
as required.
Let us choose a smooth coordinate system (x1 , x2 , . . . , xn ) over some open
set U in the smooth manifold M . Let the smooth functions Γijk on U be the
Christoffel symbols of the Levi-Civita connection on the coordinate patch U ,
defined such that
n
X
∂
∂
=
Γijk i .
∇ ∂
k
∂x
∂x
i=1
∂xj
Then Γijk = Γikj for all j and k, since
∂
∂
∂
∂
∇ ∂
−∇ ∂
=
,
= 0.
∂xk
∂xj
∂xj ∂xk
∂xj
∂xk
Let γ: I → U be a smooth curve in U , and let γ i (t) = xi ◦ γ(t) for all
t ∈ γ −1 (U ). Then
n
dγ(t) X dγ k (t) ∂
=
,
k
dt
dt
∂x
k=1
so that


n
j
X
D dγ(t)
∂ 
dγ (t)
 d γ (t) ∂ + dγ (t)
=
∇ ∂
2
k
dt dt
dt ∂x
dt j=1 dt
∂xk
k=1
j
∂x !
n
n
n
2
i
j
X d γ (t) X X
∂
dγ (t) dγ k (t)
i
=
+
Γ
.
(γ(t))
jk
2
i
dt
dt
dt
∂x
i=1
j=1 k=1
n
X
2 k
k
Thus γ: I → U is a geodesic if and only if
n
n
d2 γ i (t) X X i
dγ j (t) dγ k (t)
+
(γ(t))
=0
Γ
jk
dt2
dt
dt
j=1 k=1
(i = 1, 2, . . . , n).
Standard existence and uniqueness theorems for solutions of ordinary differential systems of equations ensure that, given a tangent vector V at any
point m of M , and given any real number t0 , there exists a unique maximal
geodesic γ: I → M , defined on some open interval I containing t0 , such that
γ(t0 ) = m and γ 0 (t0 ) = V .
198
11.2
The First Variations of Length and Energy
Definition Let M be a Riemannian or pseudo-Riemannian manifold with
metric tensor g, and let γ: [a, b] → M be a parameterized smooth curve in
M . The energy E(γ) of γ is is then defined to be the quantity
E(γ) =
1
2
Z
b
g(γ 0 (t), γ 0 (t)) dt.
a
Theorem 11.3 Let M be a Riemannian or pseudo-Riemannian manifold
with metric tensor g, let ϕ: [a, b] × (−ε, ε) → M be a smooth map, and let
γ: [a, b] → M and ϕu : [a, b] → M be the smooth curves in M defined by
γ(t) = ϕ(t, 0) and ϕu (t) = ϕ(t, u) for all t ∈ [a, b] and u ∈ (−ε, ε) (so that
γ = ϕ0 ), and let E(ϕu ) denote the energy of ϕu . Then
dE(ϕu ) = g(γ 0 (b), V (b)) − g(γ 0 (a), V (a))
du u=0
Z b D dγ(t)
−
g
, V (t) dt,
dt
dt
a
where
∂ϕ(t, u) V (t) =
.
∂u u=0
Proof The Levi-Civita connection is torsion-free. It therefore follows from
Lemma 10.2 that
D ∂ϕ(t, u)
D ∂ϕ(t, u)
=
.
du ∂t
dt ∂u
It therefore follows from Lemma 11.1 that
∂ϕ(t,
u)
∂ϕ(t,
u)
D
∂
∂ϕ(t,
u)
∂ϕ(t,
u)
1
g
,
,
= g
2
∂u
∂t
∂t
∂t
du ∂t
∂ϕ(t, u) D ∂ϕ(t, u)
= g
,
∂t
dt ∂u
Thus
dE(ϕu )
=
du
Z
b
g
a
∂ϕ(t, u) D ∂ϕ(t, u)
,
∂t
dt ∂u
and therefore
Z b dE(ϕu ) DV (t)
0
=
g γ (t),
dt
du u=0
dt
a
199
dt,
Z
b
d
(g(γ 0 (t), V (t))) dt
dt
a
Z b D dγ(t)
g
−
, V (t) dt
dt
dt
a
= g(γ 0 (b), V (b)) − g(γ 0 (a), V (a))
Z b D dγ(t)
−
g
, V (t) dt,
dt
dt
a
=
as required.
Corollary 11.4 Let M be a Riemannian or pseudo-Riemannian manifold
with metric tensor g, let ϕ: [a, b] × (−ε, ε) → M be a smooth map, and let
γ: [a, b] → M and ϕu : [a, b] → M be the smooth curves in M defined by
γ(t) = ϕ(t, 0) and ϕu (t) = ϕ(t, u) for all t ∈ [a, b] and u ∈ (−ε, ε) (so that
γ = ϕ0 ), and let E(ϕu ) denote the energy of ϕu . Suppose that ϕu (a) = γ(a)
and ϕu (b) = γ(b) for all u ∈ (−ε, ε). Suppose also that γ: [a, b] → M is a
geodesic, and thus satisfies
D dγ(t)
= 0.
dt
dt
Then
dE(ϕu ) = 0.
du u=0
Definition Let M be a Riemannian or pseudo-Riemannian manifold with
metric tensor g, and let γ: [a, b] → M be a smooth curve in M which satisfies
g(γ 0 (t), γ 0 (t)) > 0
for all t ∈ [a, b]. The length L(γ) of γ is is then defined to be the quantity
Z
L(γ) =
b
|γ 0 (t)| dt,
a
where |γ 0 (t)|2 = g(γ 0 (t), γ 0 (t)) for all t ∈ [a, b].
Theorem 11.5 Let M be a Riemannian or pseudo-Riemannian manifold
with metric tensor g, let ϕ: [a, b] × (−ε, ε) → M be a smooth map with the
property that
∂ϕ ∂ϕ
g
,
>0
∂t ∂t
200
for all t ∈ [a, b] and u ∈ (−ε, ε). Let γ: [a, b] → M and ϕu : [a, b] → M be
the smooth curves in M defined by γ(t) = ϕ(t, 0) and ϕu (t) = ϕ(t, u) for all
t ∈ [a, b] and u ∈ (−ε, ε) (so that γ = ϕ0 ), and let L(ϕu ) denote the length
of ϕu . Then
1
dL(ϕu ) 1
g(γ 0 (b), V (b)) − 0
g(γ 0 (a), V (a))
=
0
du u=0
|γ (b)|
|γ (a)|
Z b D
1 dγ(t)
g
−
, V (t) dt,
dt |γ 0 (t)| dt
a
where
∂ϕ(t, u) V (t) =
.
∂u u=0
In particular, if γ: [a, b] → M is parameterized by arclength, then
Z b D dγ(t)
dL(ϕu ) 0
0
= g(γ (b), V (b)) − g(γ (a), V (a)) −
g
, V (t) dt.
du u=0
dt
dt
a
Proof The Levi-Civita connection is torsion-free. It therefore follows from
Lemma 10.2 that
D ∂ϕ(t, u)
D ∂ϕ(t, u)
=
.
du ∂t
dt ∂u
It therefore follows from Lemma 11.1 that
2
∂ϕ(t, u) ∂ ∂ϕ(t, u) = 1 ∂ ∂ϕ(t, u) 2
∂t ∂u ∂t ∂u
∂t ∂ϕ(t, u) ∂ϕ(t, u)
1 ∂
= 2
g
,
∂u
∂t
∂t
∂ϕ(t, u) D ∂ϕ(t, u)
= g
,
∂t
du ∂t
∂ϕ(t, u) D ∂ϕ(t, u)
= g
,
∂t
dt ∂u
Thus
dL(ϕu )
=
du
Z b
∂ϕ(t, u) −1
∂ϕ(t,
u)
D
∂ϕ(t,
u)
dt,
,
∂t g
∂t
dt ∂u
a
and therefore
Z b
dL(ϕu ) 1
DV (t)
0
=
g γ (t),
dt
0
du u=0
dt
a |γ (t)|
Z b d
1
0
=
g(γ (t), V (t)) dt
|γ 0 (t)|
a dt
201
b
1 dγ(t)
, V (t) dt
−
g
|γ 0 (t)| dt
a
1
1
=
g(γ 0 (b), V (b)) − 0
g(γ 0 (a), V (a))
0
|γ (b)|
|γ (a)|
Z b D
1 dγ(t)
g
, V (t) dt,
−
dt |γ 0 (t)| dt
a
Z
D
dt
as required.
Corollary 11.6 Let M be a Riemannian or pseudo-Riemannian manifold
with metric tensor g, let ϕ: [a, b] × (−ε, ε) → M be a smooth map with the
property that
∂ϕ ∂ϕ
g
,
, >0
∂t ∂t
for all t ∈ [a, b] and u ∈ (−ε, ε). Let γ: [a, b] → M and ϕu : [a, b] → M be
the smooth curves in M defined by γ(t) = ϕ(t, 0) and ϕu (t) = ϕ(t, u) for all
t ∈ [a, b] and u ∈ (−ε, ε) (so that γ = ϕ0 ), and let L(ϕu ) denote the length
of ϕu . Suppose that ϕu (a) = γ(a) and ϕu (b) = γ(b) for all u ∈ (−ε, ε).
Suppose also that γ: [a, b] → M is a reparameterization of a geodesic, and
thus satisfies
1 dγ(t)
D
= 0.
dt |γ 0 (t)| dt
Then
11.3
dL(ϕu ) = 0.
du u=0
Jacobi Fields
Let (M, g) be a Riemannian manifold, and let γ: [a, b] → M be a geodesic
in M . A Jacobi field along γ is a vector field V along γ which satisfies the
Jacobi equation
D2 V (t)
= R(γ 0 (t), V (t))γ 0 (t),
2
dt
where R denotes the curvature tensor of the Levi-Civita connection on M .
First we show that Jacobi fields arise naturally from variations of the geodesic
γ through neighbouring geodesics.
Lemma 11.7 Let γ: I → M be a geodesic in a Riemannian manifold (M, g)
and let
ϕ: I × (−ε, ε) → M
202
be a smooth map satisfying ϕ(t, 0) = γ(t) for all t ∈ I. Let V be the vector
field along the geodesic γ defined by
∂ϕ(t, u) V (t) =
.
∂u u=0
Suppose that, for each u ∈ (−ε, ε), the curve t 7→ ϕ(t, u) is a geodesic in M .
Then the vector field V satisfies the Jacobi equation
D2 V (t)
= R(γ 0 (t), V (t))γ 0 (t).
2
dt
Proof First we note that
D ∂ϕ
= 0,
∂t ∂t
since each curve t 7→ ϕ(t, u) is a geodesic Now the Levi-Civita connection is
torsion-free. It therefore follows from Lemma 10.2 that
∂ϕ ∂ϕ ∂ϕ
D D ∂ϕ
D D ∂ϕ
−
=R
,
.
∂t ∂u ∂t
∂u ∂t ∂t
∂t ∂u ∂t
and
D ∂ϕ
D ∂ϕ
=
.
∂t ∂u
∂u ∂t
Therefore
D D ∂ϕ
D2 ∂ϕ
=
2
∂t ∂u
∂t ∂u ∂t
∂ϕ ∂ϕ ∂ϕ
D D ∂ϕ
,
+
= R
∂t ∂u ∂t
∂u ∂t ∂t
∂ϕ ∂ϕ ∂ϕ
= R
,
.
∂t ∂u ∂t
Now
∂ϕ(t, u) = γ 0 (t),
∂t u=0
∂ϕ(t, u) = V (t).
∂u u=0
Thus, on setting u = 0, we deduce that
D2 V (t)
= R(γ 0 (t), V (t))γ 0 (t),
dt2
as required.
203
11.4
The Second Variation of Energy
Let (M, g) be a Riemannian manifold and let γ: [a, b] → M be a geodesic in
M . Let ϕ: [a, b] × (−ε, ε) → M be a smooth map with the properties that
ϕ(t, 0) = γ(t) for all t ∈ [a, b],
ϕ(a, u) = γ(a) for all u ∈ (−ε, ε),
ϕ(b, u) = γ(b) for all u ∈ (−ε, ε).
Thus if ϕu : [a, b] → M is the smooth curve defined by ϕu (t) = ϕ(t, u) then
each ϕu starts at γ(a) and ends at γ(b). We calculate
d2 E(γ(ϕu )) ,
du2
u=0
where E(ϕu ) is the energy of ϕu . Let X and Y be the smooth vector fields
along the map ϕ defined by
X(t, u) =
∂ϕ(t, u)
,
∂t
Y (t, u) =
∂ϕ(t, u)
.
∂u
Note that Y (a, u) = 0 and Y (b, u) = 0 for all u ∈ (−ε, ε), on account of the
fact that ϕ(a, u) = γ(a) and ϕ(b, u) = γ(b). The energy of the curve ϕu is
given by
Z
b
E(ϕu ) =
Now
1
2
g(X(t, u), X(t, u)) dt.
a
DX
D ∂ϕ
D ∂ϕ
DY
=
=
=
∂u
∂u ∂t
∂t ∂u
∂t
by Lemma 10.2. Thus
dE(ϕu )
=
du
Z
a
b
Z b DX
DY
g X,
g X,
dt =
dt,
∂u
∂t
a
hence
Z b DX DY
D DY
g
,
+ g X,
dt
∂u ∂t
∂u ∂t
a
Z b D DY
DY DY
=
g
,
+ g X,
dt.
∂t ∂t
∂u ∂t
a
d2 E(ϕu )
=
du2
But
D DY
D DY
=
+ R(Y, X)Y
∂u ∂t
∂t ∂u
204
by Lemma 10.2. Therefore
Z b d2 E(ϕu )
DY DY
g
,
dt
=
du2
∂t ∂t
a
Z b D DY
g X,
+
+ R(Y, X)Y dt.
∂t ∂u
a
But
Z b Z b Z b D DY
∂
DX DY
DY
g X,
g
dt =
g X,
dt −
,
dt
∂t ∂u
∂u
∂t ∂u
a
a ∂t
a
DY (b, u)
DY (a, u)
= g X(b, u),
− g X(a, u),
∂u
∂u
Z b DX DY
g
−
,
dt
∂t ∂u
a
Z b DX DY
,
= −
g
dt,
∂t ∂u
a
because Y (a, u) = 0 and Y (b, u) = 0 for all u ∈ (−ε, ε). Thus
Z b DY DY
d2 E(ϕu )
=
g
,
+ g(X, R(Y, X)Y ) dt
du2
∂t ∂t
a
Z b DX DY
g
−
,
dt.
∂t ∂u
a
Now let us set u = 0. We define the vector field V along γ by
∂ϕ(t, u) V (t) = Y (t, 0) =
.
∂u u=0
Note that X(t, 0) = γ 0 (t) and
DX(t, 0)
Dγ 0 (t)
=
=0
dt
dt
(since γ is a geodesic. Therefore
d2 E(ϕu ) du2 u=0
Z b DV (t) DV (t)
0
0
=
g
,
+ g(γ (t), R(V (t), γ (t))V (t)) dt
∂t
∂t
a
Z b DV (t) DV (t)
0
0
,
+ R(γ (t), V (t), V (t), γ (t)) dt.
=
g
∂t
∂t
a
205
We can integrate the first term in this formula by parts. Using the fact that
V (a) = 0 and V (b) = 0 we see that
Z b Z b DV (t) DV (t)
D2 V (t)
g
,
dt = −
g V (t),
dt
∂t
∂t
∂t2
a
a
Also the standard properties of the Riemann curvature tensor ensure that
R(γ 0 (t), V (t), V (t), γ 0 (t)) = −R(V (t), γ 0 (t), V (t), γ 0 (t))
= R(V (t), γ 0 (t), γ 0 (t), V (t))
= g(V (t), R(γ 0 (t), V (t))γ 0 (t))
(see Proposition 9.7). We conclude that
Z b d2 E(ϕu ) D2 V (t)
0
0
g V (t), R(γ (t), V (t))γ (t) −
dt.
=
du2 u=0
dt2
a
Thus if V is a Jacobi field along γ then
d2 E(ϕu ) = 0.
du2 u=0
206