1. For a vector field X with the corresponding flow (φt )t , and a differential form ω ∈
Ω (M ), define the Lie derivative by
k
φ∗t (ω) − ω
LX (ω) = lim
.
t→0
t
For 0-forms (that is, smooth functions) this definition coincides with our earlier definition.
Prove that if X1 , . . . , Xk are vector fields, then
X
LX (ω)(X1 , . . . , Xk ) = LX (ω(X1 , . . . , Xk )) −
ω(X1 , . . . , Xi−1 , [X, Xi ], Xi+1 , . . . , Xk ).
i
Define also a map iX : Ωk (M ) → Ωk−1 (M ) by
iX (ω)(X1 , . . . , Xk−1 ) = ω(X, X1 , . . . , Xk−1 ).
In particular, iX (f ) = 0 for f ∈ C ∞ (M ). Show that
(i) iX (ω ∧ θ) = iX (ω) ∧ θ + (−1)k ω ∧ iX (θ) if ω ∈ Ωk (M );
(ii) LX (ω ∧ θ) = LX (ω) ∧ θ + ω ∧ LX (θ);
(iii) LX (f ) = iX (df ), LX (df ) = d(LX (f )), and more generally, LX = d ◦ iX + iX ◦ d.
Using the latter identity and the above formula for LX prove by induction on k that (see
p.81)
X
(dω)(X1 , . . . , Xk+1 ) =
(−1)i LXi (ω(X1 , . . . , X̂i , . . . , Xk+1 ))
i
+
X
(−1)i+j ω([Xi , Xj ], X1 , . . . , X̂i , . . . , X̂j , . . . , Xk+1 ).
i<j
2. Show that if f, g: X → S n are two continuous maps such that f (x) and g(x) are never
antipodal, then f and g are homotopic.
Show that any non-surjective continuous map into S n is homotopic to a constant map.
1
3. Let
R ω ∗be a closed 1-form on M . Show that for any smooth loop γ: S → M the
integral S 1 γ (ω) depends only on the homotopy class of γ. R
If ω = df then for any smooth curve γ: [0, 1] → M we have [0,1] γ ∗ (ω) = f (γ(1))−f (γ(0)).
Conclude that a closed 1-form ω is exact if and only if all integrals of ω along smooth loops
are zero.
4. Let Tn ∼
= Rn /Zn ∼
= (R/Z)n be the n-dimensional torus. Consider the loop γk : T → Tn ,
γk (p) = (0, . . . , 0, p, 0, . . . , 0).
k
Show that the map Ω1 (Tn ) → Rn ,
Z
ω 7→
T
γ1∗ (ω), . . . ,
Z
γn∗ (ω)
,
T
defines an isomorphism H 1 (Tn ) ∼
= Rn .
5. Let N ⊂ Rk be a smooth submanifold, U ⊃ N a tubular neighbourhood, r: U → N
the canonical projection. Show that the smooth function f (x) = kx − r(x)k on U \N satisfies
the eikonal equation
2
k X
∂f
= 1.
∂xi
i=1