PMATH 465/665: Riemannian Geometry
A useful Lemma you can use without proof for Assignment #4, problem 2[a].
Lemma. Let h C (M ) and X (T M ). Let f : M M be a dieomorphism. Then
f (hX) = (h f 1 )f X.
(1)
Note that this make sense, becaue h f 1 C (M ) and
PMATH 465/665: Riemannian Geometry
Supplementary Material on Dierentiation of the Volume Form
Let (M, g) be an oriented Riemannian manifold. We use local coordinates x1 , . . . , xn and denote the
Riemannian metric by gij = g( i , j ). The induced volume
PMATH 465/665: Riemannian Geometry
Assignment 1; due Tuesday, 24 September 2013
Note: All manifolds are always smooth in this course. Unless noted otherwise, all tensors
(functions, vector elds, forms, etc.) are also always smooth, as are all maps between
PMATH 465/665: Riemannian Geometry
Assignment 5; due Tuesday, 19 November 2013
Preamble: Recall the local coordinate formula for the Christoel symbols k of the Levi-Civita conij
nection in terms of the metric tensor:
k =
ij
1 kl
g
2
glj
gij
gli
+
xj
xi
x
PMATH 465/665: Riemannian Geometry
Assignment 4; due Tuesday, 05 November 2013
[1] Recall that in class we dened a Riemannian manifold (M, g ) to be at if it was locally isometric to
(Rn , g ), the standard n-dimensional Euclidean space.
[a] Prove that (M
PMATH 465/665: Riemannian Geometry
Assignment 3; due Tuesday, 22 October 2013
[1] Let M be a manifold with connection
(x1 , . . . , xn ), we can write
. Let be a type (k, l) tensor eld on M . In local coordinates
= i1 ik j1 jl dxi1 dxik
[a] Let X = xm b
PMATH 465/665: Riemannian Geometry
Assignment 2; due Tuesday, 08 October 2013
[1] Let G be a Lie group, and let X be a left-invariant vector eld on G. Recall that this means
(La ) X = X for any a G, where La : G G is the dieomorphism of left multiplicatio
PMATH 465/665: Riemannian Geometry
Assignment 6; due Monday, 02 December 2013
[1] Let (M, g ) be a Riemannian manifold of dimension n and let p M . Show that there exists an
open neighbourhood U of p in M and a local orthonormal frame cfw_E1 , . . . , En
PMATH 465/665: Riemannian Geometry
Assignment 1; Solutions
[1] We want to show that every vector eld X with compact support is complete. That is, we want
to show that the integral curve of X through any p M is dened for all t R. We start with a
preliminar
PMATH 465/665: Riemannian Geometry
Assignment 6; Solutions
[1] Let (M, g ) be a Riemannian manifold of dimension n and let p M . Let cfw_e1 , . . . , en be any orthonormal frame at p M . For example, we can take the coordinate frame cfw_ 1 , . . . , n o
PMATH 465/665: Riemannian Geometry
Assignment 5; Solutions
[1] Let A(p,Xp ) and B(p,Xp ) be two tangent vectors to T M at the point (p, Xp ) T M . Then we can nd
smooth curves and on T M with (0) = (0) = (p, Xp ) and (0) = A(p,Xp ) and (0) = B(p,Xp ) .
Ex
PMATH 465/665: Riemannian Geometry
Assignment 4; Solutions
[1] Recall that in class we dened a Riemannian manifold (M, g ) to be at if it was locally isometric to
(Rn , g ), the standard n-dimensional Euclidean space.
[a] Suppose that (M, g ) is at. Let p
PMATH 465/665: Riemannian Geometry
Assignment 2; Solutions
[1] Let G be a Lie group, and let X be a left-invariant vector eld on G.
[a] Let be the ow of X . Recall from Assignment #1, problem #1, we proved a lemma that said: if
there exists an > 0 such th
PMATH 465/665: Riemannian Geometry
Assignment 3; Solutions
[1] Let M be a manifold with connection
(x1 , . . . , xn ), we can write
. Let be a type (k, l) tensor eld on M . In local coordinates
.
x j1
x jl
= i1 ik j1 jl dxi1 dxik
[a] Let X = xm be a