Math 621

Spring 2015

A first course in manifolds, mainly out of Jack Lee’s Smooth Manifolds and Riemannian Manifolds, making a bee-line for the latter.

• Reading 1 (due 1/11):
  • Riemannian Manifolds Chapter 2 and see what you make of it, or
  • Smooth Manifolds Chapters 1–3 and see how far you get.
• I am away Thursday 1/15 and Tuesday 1/20.
• Homework 1 (due 2/3):
  • Write down an explicit diffeomorphism between CP¹ and S².
  • Smooth Manifolds problems 2-10 and 3-4.
  • Show that if X and Y are topological spaces, π: E → Y is a vector bundle, and f: X → Y is continuous, then f ^*E is naturally a vector bundle over X. If the definition of vector bundle given in class was too vague for you, have a look at chapter 10 of Smooth Manifolds.
• Reading 2 (due 2/15): Riemannian Manifolds, Chapters 3 and 4.
• Homework 2 (due 2/24): Smooth Manifolds problems 4-6, 8-13 (give an explicit formula for the vector field), 9-3 (choose one part), 9-7, 11-4, and 16-2.
• Reading 3 (due 3/8): Riemannian Manifolds, Chapters 5 and 6.
• Homework 3 (due 3/19): pdf tex.
• Make-up lecture: Monday 4/6, 10:20–11:10, Physics 299.
• I am away Tuesday 4/7 and Thursday 4/9.
• We will meet at the regular time and place on 4/14, 14/16, and 4/21.
• Homework 4 (due 6/2):
  • Lee defines the exponential map on a Lie group G as follows: Given a tangent vector X ∈ T_I(G), extend it to a left-invariant vector field, integrate it to get a flow, and let exp(X) be the image of I at time 1. Show that if G = GL_n(R) then this agrees with the usual (matrix) exp. Optional: Show the same if G is a Lie subgroup of GL_n(R).
  • Riemannian Manifolds #3-12b, and as many of the preceding problems as you need to make it work.
  • Consider S³ as the unit sphere in C², and the map from SU(2) to S³ sending a matrix to its first column. Argue (briefly but explicitly) that this is a diffeomorphism. Show that the pull-back of the round metric is a bi-invariant metric.
  • Riemannian Manifolds #5-11.
  • Riemannian Manifolds #7-5.
  • If M is a compact Riemannian manifold then for any p ∈ M, the exponential map T_p(M) ← M is surjective. This follows from the Hopf–Rinow theorem, which Lee proves in chapter 6; there is also a nice proof in chapter 10 of Milnor’s Morse theory. In contrast, show that if G = GL₂⁺(R) then the (matrix) exponential map is not surjective, and in particular that

    -1	1
    0	-1

    is not in the image. Hint: Use Jordan normal form. Optional: Describe the image of the exponential map completely.