Geometry & Topology Toolkit
Guo Jincheng
November 5, 2022
Introduction
Differential Geometry & Symplectic Geometry
i ...i
Formula 0.1 (Transformation Rule of Tensors). Tj11...jqp are the coefficients of coordinates x1 , . . . , xn :
X
i ...i
Tj11...jqp =
(k),(l)
i1
∂xip ∂z l1
∂z lq
k ...k ∂x
Tel11...lq p k1 · · · kp j1 · · · jq .
∂z
∂z ∂x
∂x
Formula 0.2 (Lie Derivative).
i ...i
Lξ Tj11...jqp
d
i ...i
=
(Ft T )j11 ...jpq
dt
where
l ...l
i ...i
(Ft T )j11 ...jpq = Tk11 ...kpq
∂xk1
∂xj01
(1)
(2)
t=0
i
···
∂xkq ∂xi01
∂x p
· · · 0lp
jq ∂xl1
∂x
∂x0
(3)
and Ft is the local one-parameter group;
i i
p
∂Tj11...j
q
∂ξk
∂ξ k
i1 ...ip
+
·
·
·
+
T
j
...j
1
q−1k
j
j2 ......jq ∂x 1
∂xs
∂xjq
ip
i1
i ...i ∂ξ
l ...i ∂ξ
− · · · − Tj11...jqp
.
− Tj11...jqp
∂xl
∂xl
i ...i
Lξ Tj11...jqp
=ξ
s
+ Tki1 ...ip
(4)
More generally, for vector field Y ,
(LX Y )m
dX−t YXt (m) − Ym
d
= lim
=
t→0
t
dt
dX−t YXt (m)
(5)
t=0
and for a differential form ω,
(LX ω)m
δXt ωXt (m) − ωm
d
= lim
=
t→0
t
dt
δXt ωXt (m)
.
(6)
t=0
If T is a tensor field of type (r, s), then (LX T )m is the derivative at t = 0 of the (Mm )(r,s) -valued function
whose value at t is
dX−t (v1 ⊗ · · · ⊗ vr ) ⊗ δXt (v1∗ ⊗ · · · ⊗ vs∗ )
if
T |Xt (m) = v1 ⊗ · · · ⊗ vr ⊗ v1∗ ⊗ · · · ⊗ vs∗ .
(7)
Formula 0.3 (Cartan’s Formula).
(k + 1)dω (X1 , . . . , Xk+1 )
X
=
(−1)i ∂Xi ω X1 , . . . , X̂i , . . . , Xk+1
i
+
X
(−1)
i+j
ω [Xi , Xj ] , X1 , . . . , Ẋi , . . . , Ẋj , . . . , Xk+1
i<j
1
(8)
Alternatively, let Yi be smooth vector fields,
dω (Y0 , . . . , Yp ) =
p
X
(−1)i Yi ω Y0 , . . . , Ybi , . . . , Yp
i=0
+
X
(9)
(−1)i+j ω [Yi , Yj ] , Y0 , . . . , Ybi , . . . , Ybj , . . . , Yp .
i<j
More generally:
LX = i(X) ◦ d + d ◦ i(X).
(10)
Formula 0.4 (Covariant Differentiation). The transformation rule of the Christoffel Symbols
Γksq = −
is
′
′
Γkp′ q′ =
∂z k
∂z k
∂xi ∂xm ∂ 2 z k
∂z s ∂z q ∂xi ∂xm
Γkpq
∂z p ∂z q
∂2zk
′
′ +
p
q
∂z ∂z
∂z p′ ∂z q′
(11)
.
Then the formula for connection under coordinate z is given by
∂ Te(l)
(k)
(k)
Te(l);r =
∂xr
+
p
X
k ...(k →i)...kp ks
Tel11...lq s
Γir −
s=1
q
X
k ...k
Tel11...(lsp→i)...lq Γils r .
(12)
s=1
Conventionally,
(i)
Also,
(i)
∇k T(j) = T(j):k .
(13)
(j)
(j)
(i)
(i)(j)
(i)
∇k T(p)(q) = ∇k R(p) S(q) + R(p) ∇k S(q) .
(14)
The operation of covariant differentiation commutes with contractions.
Also we define the directional derivative along vector ξ as
(i)
(i)
∇ξ T(j) = ξ k ∇k T(j) .
(15)
∇T (T ) = 0.
(16)
Formula 0.5 (Geodesic).
In component,
d2 xj
dxk dxi
= 0,
+ Γjki
2
dt
dt dt
(17)
j = 1, . . . , n.
Formula 0.6 (Christoffel’s Formula). For a given non-degenerate metric gij , we can find a compatible and
symmetric connection using
1
∂glj
∂gil
∂gij
Γkij = g kl
+
−
.
(18)
2
∂xi
∂xj
∂xl
Theorem 0.1 (Existence of Connection Compatible with a Hermitian Metric). Given a hermitian metric
ds2 = hik̄ dz i dz̄ k ,
(19)
hkī = h̄ik̄
k
on a region D ⊂ Cn , and let DR denote its realized space where ds2R = Re (hik̄ ) dxi dxk + dy i dy .
The symmetric connection on DR compatible with ds2R is compatible with ds2 if and only if the form
Ω=
i
h dz j ∧ dz̄ k
2 j k̄
(20)
is closed. Or equivalently, the latter metric is Kaehlerian.
Formula 0.7 (Curvature Tensor).
p
i
(∇k ∇l − ∇l ∇k ) T i = −Rqkl
T q + Tkl
∂T i
,
∂xp
(21)
p
where Tkl
is the torsion tensor. If the connection is symmetric,
i
(∇k ∇l − ∇l ∇k ) T i = −Rqkl
T q,
2
(22)
where
∂Γiql
∂Γiqk
−
+ Γipk Γpql − Γipl Γpqk .
∂xk
∂xl
Moreover, the curvature tensor enjoys the following symmetries:
i
−Rqkl
=
i
i
Rqkl
= −Rq!k
,
(23)
(24)
and for connection compatible with the metric g,
Riqkl = −Rqikl ,
(25)
i
i
i
Rqkl
+ Rklq
+ Rlqk
= 0,
(26)
and if the connection is symmetric,
when the connection is both symmetric and compatible with the metric,
Riqkl = Rkliq .
(27)
Also, we define the contraction of the curvature tensor
i
Rql = Rqil
(28)
i
R = g lq Rql = g lq Rqil
(29)
as the Ricci tensor and define
as the scalar curvature with metric g. Therefore we can deduce the Gauss’ ”Theorema Egregium”.
Topology
Geometry and Physics
3