MA40254 Differential and geometric analysis : Exercises 9
Hand in answers by 1:15pm on Wednesday 6 December for the Seminar of Thursday 7 December
Homepage: http://moodle.bath.ac.uk/course/view.php?id=57709
0 (Warmup). Let φ : R2 → R2 ; p 7→ (x1 (p), x2 (p)2 ). Compute φ∗ (dx1 ∧ dx2 ) directly, and find the
largest open subset of R2 on which φ is an orientation-preserving local diffeomorphism.
[Solution: φ∗ (dx1 ∧ dx2 ) = d(φ∗ x1 ) ∧ d(φ∗ x2 ) = dx1 ∧ d(x22 ) = 2x2 dx1 ∧ dx2 , which is Jφ dx1 ∧ dx2
in accordance with the general result in lectures, since the matrix of Dφp is diagonal with entries
1, 2x2 (p) hence Jφ = 2x2 . Thus φ is a local diffeomorphism when x2 ̸= 0 and orientation preserving
on {p ∈ R2 : x2 (p) > 0}.]
1. Let M ⊆ Rs be an orientable submanifold, and let U ⊆ M be an open subset. Show that U is also
orientable.
[Hint: If ω ∈ Ωn (M ) is an orientation form, what can you say about its pullback to U ?]
2 (Less essential). Let M ⊆ Rn+1 be a submanifold of dimension n. Show that M is orientable if and
only if there is a nowhere-vanishing normal vector field on M , i.e., a smooth function ν : M → Rn+1
such that ν(p) ̸= 0 and ν(p) is orthogonal to Tp M for all p ∈ M .
[Hint: Given such a ν, consider ω ∈ Ωn (M ) defined by ωp = ν(p) ⌟ Det. Show that ν is uniquely
determined by ω—you may then assume that ν is smooth (this is not so easy to prove rigorously).]
3. Let S n = {x ∈ Rn+1 : ∥x∥ = 1}, and let a : S n → S n be the antipodal map, i.e., the diffeomorphism
p 7→ −p. For which values of n is a orientation-preserving?
[Hint: If ω ∈ Ωn (S n ) is an orientation form, then a∗ ω = f ω for some function f : S n → R \ {0}. You
need to decide whether f takes positive or negative values. Consider the orientation form ω ∈ Ωn (S n )
given by ωp = p ⌟ Det.]
4. Let U and Ũ be open subsets of Rn , and α ∈ Ωncpt (U ). Let φ : Ũ → U be an orientation-reversing
diffeomorphism, i.e., det(Dφp ) < 0 for all x ∈ Ũ . Show that
Z Z
φ∗ α = − α.
Ũ U
[Hint: Imitate the proof of that the integral is invariant under orientation-preserving diffeomorphisms.]
5. Plan an essay on one of the following topics.
(i) The inverse function theorem and its use in submanifold theory.
(ii) Alternating multilinear forms and their properties.
(iii) Using pullback to define the exterior derivative on submanifolds.
MA40254 Differential and geometric analysis : Solutions 9
1. Let ω ∈ Ωn (M ) be an orientation form, so that i∗ ω ∈ Ωn (U ), with i : U → M the inclusion. Then
for any p ∈ U , (i∗ ω)p = ωp ∈ Altn (Tp U ) is non-zero, so i∗ ω is an orientation form too.
2. For a fixed p ∈ M , consider Np ⊆ Rk+1 the vector space orthogonal to Tp M . Then Np has
dimension 1, and the map
Np → Altn (Tp M ), u 7→ (u ⌟ Det)|Tp M n
is an isomorphism. Thus for each p ∈ M any ωp ∈ Altn (Tp M ) can be written uniquely as ωp =
ν(p) ⌟ Det for some ν(p) ∈ Np . Clearly if ν : M → Rn+1 has smooth local extensions, so does ω (using
the same formula) and in fact the converse is also true (proof omitted). Now ωp ̸= 0 if and only
ν(p) ̸= 0, so ω is an orientation form if and only if ν never vanishes.
3. Define ω ∈ Ωn (S n ) by ωp = p ⌟ Det. Then x is a normal vector field to S n and so ω is an orientation
form.
Now compare ω and a∗ ω. First note that ω−p = (−p) ⌟ Det = −(p ⌟ Det) = −ωp ; that we can at
all compare ωp and ω−p like this relies on Tp S n = T−p S n . Meanwhile Dap = − IdTp S n for any p ∈ S n .
Therefore
(a∗ ω)p = (Dap )∗ (ω−p ) = (− Id)∗ (−ωp ) = −(det(− Id))ωp = (−1)n+1 ωp ∈ altn (Tp S n ).
So a∗ ω = (−1)n+1 ω, implying that a is orientation-preserving if and only if n is odd.
4. Write α = f dy1 ∧ · · · ∧ dyn . Since |J(x)| = −J(x),
Z Z Z
∗
φ α= f (φ(x))J(x)dx1 · · · dxn = − f (φ(x))|J(x)| dx1 · · · dxn
Ũ ŨZ Z Ũ
=− f (y)dy1 · · · dyn = − α.
U U
5. We do not give model essays, and there is some flexibility on what you cover (in particular, what
you prove and in what level of detail, and what examples you give). Here are some comments and
suggestions.
(i) You certainly need to state the inverse function theorem and the regular value theorem, which
means defining what is a submanifold and what is a regular value. A sketch proof of the inverse
function theorem would be too long, but you might explain “why” the inverse function theorem
is true at a higher level, and sketch how the regular value theorem follows from it. You should
also give an example of a submanifold defined as the inverse image of a regular value.
(ii) Define alternation and wedge, state what is a basis, prove something, and give an example
computation (e.g., you could discuss decomposability).
(iii) To define the exterior derivative on submanifolds, you will need the result that exterior derivative
commutes with pullback, and to define what is a differential form on a submanifold. A proof
and an example should be easy to come by.
