exercise sheet - CMUP

Differentiable Manifolds
Exercises
1. For i = 1, 2, consider the charts φ i : R → R defined by φ 1 (x) = x and φ 2 (x) = x 3 .
(a) Show that φ 1 and φ 2 define different smooth structures on the topological manifold R.
(b) Let M i denote the smooth manifold defined by the chart φ i for i = 1, 2. Show that M 1
and M 2 are diffeomorphic.
(c) Is it possible to find a smooth structure on R n such that the resulting smooth manifold
M is not diffeomorphic to the manifold R n with the standard smooth structure (coming
from the identity chart Id: R n → R n )?
2. Let F : V → W be a linear map between finite dimensional vector spaces and let A be the
matrix of F with respect to given bases of V and W . Show that the transpose A t is the matrix
of F ∗ : W ∗ → V ∗ with respect to the dual bases of the given ones.
3. Show that the map det: GL(n, R) → R is a smooth map and that its derivative at I ∈
GL(n, R) is given by
det ∗ (I)(X) = tr X
for X ∈ M n (R) ∼ = T I GL(n, R).
4. Let SL(n, R) = {A ∈ GL(n, R) : det(A) = 1}.
(a) Show that SL(n, R) is a submanifold of GL(n, R) and find its dimension.
(b) Determine the tangent space T I SL(n, R) as a subspace of T I GL(n, R) ∼ = M n (R).
5. Consider the standard charts on RP n defined by using homogeneous coordinates:
φ −1
i
: R n → RP n ,
(x 0 , . . . , ̂x i , . . . , x n ) ↦→ [x 0 : · · · : 1
i
: · · · : x n ],
where the notation ̂x i indicates that the coordinate x i is omitted.
(a) Calculate explicitly the coordinate transformations φ i ◦ φ −1
j
.
(b) Let U i = φ −1
i
(R n ) = {[x] : x i ≠ 0} ⊂ RP n . Show that the complement RP n U i is a
smooth submanifold diffeomorphic to RP n−1
(c) Let l i : a i x + b i y + c i = 0, i = 1, 2 be distinct lines in R 2 and let ¯l i be their closures in RP 2
under the embedding R ↩ → RP 2 given by (x, y) ↦→ [x : y : 1]. Find the defining equations
for ¯l 1 and ¯l 2 in RP 2 and determine their point of intersection.
6. (a) Let U and V be finite dimensional vector spaces of dimension k and n − k respectively.
Show that there is a bijection
Hom(U, V ) ∼ = { k-dimensional subspaces Γ ⊂ U ⊕ V such that Γ ∩ V = {0} } ,
where Hom(U, V ) ∼ = R k(n−k) denotes the set of linear maps f : U → V .
(b) Let G k (R n ) denote the set of k-dimensional linear subspaces of R n . Use the previous
question to define charts on G k (R n ) making it into a smooth manifold of dimension
k(n − k).

7. Let N ↩→ M be an embedding such that N is closed in M and let f : N → R be a smooth
function. Show that there exists a smooth function g : M → R such that g |N = f.
Hint: partition of unity.
8. Let M be a smooth compact one-dimensional manifold. What can you say about it? How
many examples of such manifolds can you come up with?
9. Let X(M) the space of smooth vector fields on the manifold M and let Der(C ∞ (M)) be
the space of derivations of the R-algebra C ∞ (M), i.e., R-linear maps D : C ∞ (M) → C ∞ (M)
satisfying D(fg) = D(f)g + fD(g). Show that the map
Ψ: X(M) → Der(C ∞ (M))
defined by
is an isomorphism.
Ψ(X)(f) = X(f)
10. Let U ⊂ R m be open and let X, Y ∈ X(U) be vector fields given by
X = a i
∂
, a i ∈ C ∞ ∂
(U) and Y = b i ,
∂x i ∂x i
b i ∈ C ∞ (U).
Show that the bracket [X, Y ] ∈ X(U) is given by
[X, Y ] =
(
)
∂b j ∂a j
a i − b i
∂x i ∂x i
∂
∂x j
.
11. Let M be a smooth manifold and let X, Y ∈ X(M) be smooth vector fields. Show that
for any smooth functions f, g ∈ C ∞ (M).
[fX, gY ] = fg[X, Y ] + (fXg)Y − (gY f)X
12. Consider the canonical identification of the tangent space of GL(n, R) at the identity with
the space of all n × n-matrices:
T I GL(n, R) = M n (R)
and let [X, Y ] = XY − Y X be the commutator of matrices. Consider the linear map
given by
Ψ: T I GL(n, R) → X(GL(n, R))
Ψ(X)(P ) = ˜X(P ) = P X
for X ∈ T I GL(n, R) and P ∈ GL(n, R), and where P X denotes the product of matrices.
Show that
Ψ([X, Y ]) = [Ψ(X), Ψ(Y )]
Hint: You may want to use the bracket of vector fields in local coordinates from above.
Remark: Give T I GL(n, R) the Lie algebra structure from the commutator of matrices and give
X(GL(n, R)) the Lie algebra structure from the bracket of vector fields. Then you are asked to
show that Ψ is a homomorphism of Lie algebras.

13. Consider the chart φ on R 2 {0} given by (x, y) = φ −1 (r, θ) = (r cos θ, r sin θ) for (r, θ) ∈
R + ×] − π, π[. Let ω ∈ Ω 1 (R 2 {0}) be given by
ω(x, y) =
−y dx + x dy
x 2 + y 2 .
Calculate the expression for ω in the local coordinates (r, θ).
14. Let U ⊂ R 3 be open. Show that there is a commutative diagram
C ∞ (U)
⏐
∇
−−−−→ X(U)
∇×
−−−−→ X(U)
∇·
−−−−→ C ∞ (U)
⏐
↓φ 0
⏐ ⏐↓
φ 1
⏐ ⏐↓
φ 2
⏐ ⏐↓
φ 3
Ω 0 (U)
d
−−−−→ Ω 1 (U)
d
−−−−→ Ω 2 (U)
where the isomorphisms φ i , i = 0, 1, 2, 3 are defined by
φ 0 = Id,
φ 1 (f 1 , f 2 , f 3 ) = f 1 dx 1 + f 2 dx 2 + f 3 dx 3 ,
d
−−−−→ Ω 3 (U),
φ 2 (g 1 , g 2 , g 3 ) = g 1 dx 2 ∧ dx 3 + g 2 dx 3 ∧ dx 1 + g 3 dx 1 ∧ dx 2 ,
φ 3 (h) = h dx 1 ∧ dx 2 ∧ dx 3 .
15. Let θ : U → V be a diffeomorphism between open sets U, V ⊂ R m . Let X ∈ X(U) be a
vector field and write
∂
X = a i , a i ∈ C ∞ (U).
∂x i
Find the smooth functions b i ∈ C ∞ (V ) such that
θ ∗ X = b j
∂
∂y j
.
16. Let θ : U → V be a diffeomorphism between open sets U, V ⊂ R m . Let ω ∈ Ω 1 (V ) be a
1-form and write
ω = a i dx i , a i ∈ C ∞ (V ).
Find the smooth functions b i ∈ C ∞ (U) such that
17. Let ω ∈ Ω 1 (R 2 {(0, 0)} be defined by
ω(x, y) =
θ ∗ ω = b j dx j .
−y dx + x dy
x 2 + y 2 ,
and let f : R → R 2 be the smooth map f(t) = (cos t, sin t). Calculate the pull-back
f ∗ ω ∈ Ω 1 (R).
18. Consider the real projective space RP n with homogeneous coordinates
and let, as usual,
[x] = [x 0 : · · · : x n ] ∈ RP n
U i = {[x 0 : · · · : x n ] ∈ RP n : x i ≠ 0}.
Recall the definition of the tautological line bundle H p −→ RP n :
Consider the trivializations
H = {([x], v) ∈ RP n × R n+1
φ i : U i × R → H |Ui = p −1 (U i )
: ∃λ ∈ R such that v = λx}.
([x], t) ↦→ ( [x], (t/x i ) · x ) .

(a) Find the corresponding transition functions g ij ([x]) for H.
(b) Find the transition functions for the dual bundle H ∗ → RP n .
(c) Find the transition functions for the tensor powers 1
H ⊗k → RP n , k ∈ N.
19. Let E → M and F → M be vector bundles with trivializations φ α for E and ψ α for F over
open sets U α , such that {U α } is an open covering of M. Find transition functions which define
a vector bundle Hom(E, F ) → M with the property that there is a natural identification
Hom(E, F ) x
∼ = Hom(Ex , F x ).
20. Let H ⊥ → RP n be the rank n vector bundle defined by
H ⊥ = {([x], w) ∈ RP n × R n+1 : 〈x, w〉 = 0}.
(a) Show that the Whitney sum H ⊕ H ⊥ is isomorphic to the trivial vector bundle RP n ×
R n+1 → RP n of rank n + 1 over RP n .
(b) (Challenge!) Show that the tangent bundle T RP n is isomorphic to the vector bundle
Hom(H, H ⊥ ).
21. Let [a, b] ⊂ R be a closed interval. Let ω ∈ Ω 1 ([a, b]) be a smooth 1-form 2 : this can be
written
ω(t) = g(t)dt
for some smooth function g on [a, b]. Define
∫
ω =
[a,b]
∫ b
a
g(t)dt.
(a) Let r : [c, d] → [a, b] be a smooth map such that r(c) = a and r(d) = b. Calculate r ∗ ω and
show that
∫ ∫
r ∗ ω = ω.
[c,d]
Let now M be a smooth manifold and let ω ∈ Ω 1 (M) be a smooth 1-form. For any smooth
curve γ : [a, b] → M, define
∫ ∫ b
ω = γ ∗ ω.
γ
(b) Let ˜γ(s) = γ(r(s)) be a reparametrization of γ (where r is as above). Show that
∫ ∫
ω = ω.
γ
(c) Assume that ω = df for some smooth function f : M → R (we say that ω is exact). Show
that
∫
ω = f(γ(b)) − f(γ(a)).
[a, b].
γ
a
˜γ
[a,b]
In particular, if γ is a closed curve, then ∫ γ
df = 0.
1 This exercise is for those who know tensor products.
2 By this we mean that ω is the restriction to [a, b] of a smooth 1-form defined on some open interval containing

(d) Let ω ∈ Ω 1 (R 2 {(0, 0)}) be defined by
ω(x, y) =
−y dx + x dy
x 2 + y 2 ,
and let γ : [0, 2π] → R 2 {(0, 0)} be the curve γ(t) = (cos t, sin t). Calculate
∫
ω
and deduce that ω is not exact.
22. Let T 2 = S 1 × S 1 be the two-torus. Let
U = T 2 (S 1 × {(0, 1)}),
V = T 2 (S 1 × {(0, −1)}).
Use the Mayer–Vietoris sequence for T 2 = U ∪ V to calculate the de Rham cohomology of T 2 .
23. Let M be any smooth manifold of dimension m and let p ∈ M.
(a) Show that there exists a smooth map
γ
f : M → S m
which is a local diffeomorphism at p. Hint: Recall that the quotient space D m /∂D m is
homeomorphic to S m , where D m ⊂ R m is the closed m-disk. Use a chart on M to first
find a continuous map with the required property and then apply exercise 6.1 of Barden
& Thomas.
(b) Let f : M → S m be a smooth map as in (a) and let j : M {p} ↩→ M be the inclusion.
Show that the composition
is the zero map.
j ∗ ◦ f ∗ : H m (S m ) → H m (M {p})
(c) Let f : M → S m be a smooth map as in (a). If M is connected, show that
is surjective. Deduce that
is the zero map.
f ∗ : H m (S m ) → H m (M)
j ∗ : H m (M) → H m (M {p})
(d) Let M be a connected smooth manifold and let p ∈ M. Let f : M → S m be a smooth
map as in (a) and let U ⊂ M be an open neighbourhood of p diffeomorphic to an open
disc in R m , such that f |U : U → S m is a diffeomorphism onto its image.
Let V = M {p} and use the Mayer–Vietoris sequence for the covering M = U ∪ V to
show that H m (M {p}) = 0.
24. Let S g be a compact smooth oriented surface without boundary of genus g 0. Use an
inductive argument based on the Mayer–Vietoris sequence and exercises 6.4 and 6.5 of Barden
& Thomas to calculate H ∗ (S g ).

25. Complex projective n-space is defined by
CP n = (C n {0})/ ∼ where z ∼ w if z = λw for some λ ∈ C {0}.
As for RP n , write [z] = [z 0 : · · · : z n ] for z = (z 0 , . . . , z n ) ∈ C n {0}.
(a) Let U i = {[z] ∈ CP n
: z i ≠ 0}. Show that the charts
φ i : U i → C n
[z 0 : · · · : z n ] ↦→ z z i
make CP n into a compact connected oriented smooth manifold (use the standard identification
C n ∼ = R 2n given by (z 1 , . . . , z n ) ↦→ (x 1 , y 1 , . . . , x n , y n ), where for z j = x j + iy j ,
and the standard orientation dx 1 ∧ dy 1 ∧ · · · ∧ dx n ∧ dy n ).
(b) Let V i := CP n {[0 : · · · : 0 : 1 : 0 · · · : 0]} . Show that the map (omitting the ith coordinate)
is a homotopy equivalence.
i
V i → CP n−1
[z] ↦→ [z 0 : · · · : ẑ i : · · · : z n ]
(c) Use the Mayer–Vietoris sequence for the covering CP n = U n ∪V n to show that the inclusion
induces an isomorphism
j : CP n−1 ↩→ CP n
[z 0 : · · · : z n−1 ] ↦→ [z 0 : · · · : z n−1 : 0]
j ∗ : H k (CP n ) ∼ = −→ H k (CP n−1 ) for 1 < k < 2n − 1.
(d) Show that H 2k (CP n ) ∼ = R and H 2k+1 (CP n ) ∼ = 0 for 0 k n.