MAT 367S – Assignment #4
Due in class on Friday, March 6, 2015
Problem #1: [5 points]
Let F : M → N be a submersion. Show that for any submanifold S ⊂ N , the pre-image F −1 (S) ⊂ M
is a submanifold. (Hint: Use the normal form for submersions.)
Problem #2: [5 points]
Let S ⊂ R3 be a 2-dimensional submanifold, and let f : S → R be the map f (x, y, z) = z. Show that
p ∈ S is a critical point for f if and only if the tangent space Tp S ⊂ R3 is the x-y plane. (Hint: View
f as the restriction of a function g : R3 → R, (x, y, z) 7→ z.)
Problem #3: [5 points]
For any complex matrix A ∈ Matn (C), let A¯ be its complex conjugate, and A† = A¯> the conjugate
transpose. Let
Herm(n) = {A ∈ Matn (C)| A† = A},
U(n) = {U ∈ Matn (C)| U † U = In }
be the real subspace of Hermitian matrices, and the group of unitary matrices. By studying the map
F : Matn (C) → Herm(n), A 7→ A† A
2
2
show that the group U(n) is a submanifold of Matn (C) ∼
= Cn = R2n , and describe its tangent space
at In ∈ U(n). What is the dimension? (Hint: Compare with our discussions of O(n) ⊂ Matn (R), from
the last problem set.)
Problem #4: [5 points]
Let S 2 ⊂ R3 be the 2-sphere, and define
T S 2 = {(p, v) ∈ R3 × R3 | p ∈ S 2 , v ∈ Tp S 2 }.
It may be regarded as a level set of the function F : R6 → R2 , F (p, v) = (p · p, p · v).
a) Find the differential (a, w) 7→ (T(p,v) F )(a, w).
b) Show that T S 2 is a 4-dimensional submanifold of R6 .
c) Show similarly that M = {(p, v) ∈ T S 2 | v · v = 1} is a 3-dimensional submanifold of R6 .
Note: The manifold M has interesting properties: Both maps (p, v) 7→ p, (p, v) 7→ v are surjective
submersions M → S 2 , with fibers diffeomorphic to S 1 ; the map (p, v) 7→ (v, p) restricts to a diffeomorphism of M interchanging these two maps.
Continued on back side.
Extra question (Do not hand in.)
Let J ∈ Mat2n (R) be the matrix, written in block form as
0n In
J=
−In 0n
where In is the n × n identity matrix, and 0n is the n × n zero matrix.
The set Sp(2n) ⊂ Mat2n (R) consisting of matrices A with the property
A> JA = J
is a group. Show that Sp(2n) is a submanifold of Mat2n (R), and describe the tangent space at
I2n ∈ Sp(2n). What is the dimension of Sp(2n)?
Extra question (Do not hand in.)
Consider again the manifold M ⊂ T S 2 , given as the unit tangent bundle. Let π : M → S 2 , (p, v) 7→ p.
Let (U± , φ± ) be the usual charts given by stereographic projection. Describe the maps
T φ± : T U± → R2 × R2 .
Let F± : π −1 (U± ) → R2 × S 1 be the restriction. Find the map
F+ ◦ (F− )−1 : R2 − {0} × S 1 → R2 − {0} × S 1 .
Compare with the results for the Hopf fibration S 3 → S 2 .