Math 660, Homework 1
Review of smooth manifolds
due Thursday, September 15
Reading do Carmo, chapter 0
Exercises (to do on your own)
1. Prove that if M and N are smooth manifolds, then the product space M N
becomes a smooth manifold in a natural way. Also
Math 660, Homework 9
Hadamards Theorem, Spaces of Constant Curvature
due November 10, 2011
Reading do Carmo, chapter 7, 8.18.4
Exercises (to do on your own)
1. Find a local dieomorphism f : M N that is not a covering map. Also, put a
complete Riemannian m
Math 660, Homework 8
GaussBonnet, complete manifolds
due November 3, 2011
Reading do Carmo, chapter 7
Exercises (to do on your own)
1. Let (M, g ) be a compact, oriented Riemannian manifold with nonempty boundary
M . Using the GaussBonnet formula and assu
Math 660, Homework 7
Submanifolds
due October 27, 2011
Reading do Carmo, chapter 6
Exercises (to do on your own)
1. Check that B (X, Y ) is well-dened (independent of the extensions of X and Y ).
2. Derive Theorem 2.5 on pg. 130 from the Gauss equation (P
Math 660, Homework 5
Jacobi elds
due October 13, 2011
Reading do Carmo, chapter 5
Exercises (to do on your own)
1. Verify any unproved results from lecture.
2. Consider the standard cylinder R S 1 with the product metric. What does the
conjugate locus of
Math 660, Homework 5
Curvature: parts I and II
due October 6, 2011
Reading do Carmo, chapter 4
Exercises (to do on your own)
l
1. Derive the expression for Rijk in terms of the Christoel symbols (formula (2) on
pg. 93 of do Carmo. (Note that this formula
Math 660, Homework 4
Geodesics: parts I and II
due September 29, 2011
Reading do Carmo, chapter 3
Exercises (to do on your own)
1. Let : (M, g ) (N, h) be an isometry, and suppose : I M is a geodesic in
M . Prove that is a geodesic in M .
2. Suppose that
Math 660, Homework 3
Connections
due Thursday, September 22
Reading do Carmo, chapter 2
Exercises (to do on your own) Let M be a smooth manifold with connections
, and Riemannian metric , .
1. Explain why
c + (1 c)
1
2
and +
are not connections. On the ot
Math 660, Homework 2
Riemannian metrics
due Thursday, September 22
Reading do Carmo, chapter 1
Exercises (to do on your own) Except where indicated, (M, g ) is a Riemannian nmanifold.
1. Prove that Isom(M, g ), the set of isometries of (M, g ), is a group
Math 660, Homework 10
BonnetMyers, SyngeWeinstein, Rauch, Morse Index Theorem
due Thursday December 1st, 2011
Reading do Carmo, chapter 9, 10.110.2, 11
Exercises (to do on your own)
1. Describe a non-constant geodesic in a manifold that is non-minimizing,