Mathematics G4404. Modern Geometry
Fall 2014
Assignment 10
Due Wednesday, December 3, 2014
(1) (2nd Bianchi identity) Let (M, g) be a Riemannian manifold. Prove
that
R(X, Y, Z, W, T ) +
R(X, Y, W, T, Z) +
R(X, Y, T, Z, W ) = 0
for all X, Y, Z, W, T X (M )
Mathematics G4402. Modern Geometry
Fall 2014
Assignment 10
Due Monday, November 24, 2014
(1) (geodesic frame) Let (M, g) be a Riemannian manifold of dimension n and let p M . Show that there exists an open neighborhood U M of p and n vector elds E1 , . .
Mathematics G4402. Modern Geometry
Fall 2014
Assignment 8
Due Monday, November 10, 2014
(1) Let H = cfw_(y1 , y2 ) R2 | y2 > 0 be the upper half plane, and dene
a Riemannian metric on H by
g=
2
2
dy1 + dy2
.
2
y2
(a) Compute the Christoel symbols k , i, j
Mathematics G4402. Modern Geometry
Fall 2014
Assignment 9
Due Monday, November 17, 2014
(1) Let (a, b) T S n = cfw_(x, y) Rn+1 Rn+1 | |x| = 1, x y = 0.
Assume that b = 0. Prove that
: R S n,
t cos(|b|t)a + sin(|b|t)
b
|b|
is the unique geodesic on (S n ,
Mathematics G4402. Modern Geometry
Fall 2014
Assignment 5
Due on Monday, October 13, 2014
[GHL] = S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Third Edition
(1) Dene a smoooth function Q on Rn+1 by
Q(x0 , x1 , . . . , xn ) = x2 + x2 + + x2 .
0
Mathematics G4402. Modern Geometry
Fall 2014
Assignment 7
Due Wednesday, October 29, 2014
(1) Leg G be a Lie group, and let g = Te G be its Lie algebra, where e is
the identity element of G. Dene i : G G by i(x) = x1 . Prove
the following statements.
(a)
Mathematics G4402. Modern Geometry
Fall 2014
Assignment 6
Due on Monday, October 20, 2014
[dC]= do Carmo, Riemannian Geometry
(1) Let G M M be a properly discontinuous action of a group
G on a smooth manifold M . Let g , g G, and the smooth
manifold M/G,
Mathematics G4402. Modern Geometry
Fall 2014
Assignment 2
Due on Monday, September 22, 2014
(1) Prove that the tangent bundle T M of a smooth manifold M is
an orientable manifold (even though M may not be).
(2) Let p(x1 , . . . , xk ) R[x1 , . . . , xk ]
Mathematics G4402. Modern Geometry
Fall 2014
Assignment 4
Due on Monday, October 6, 2014
[GHL] = S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Third Edition
(You can nd the pages in [GHL] needed for this assignment at http:/www.amazon.com)
(1)
Mathematics G4402. Modern Geometry
Fall 2014
Assignment 3
Due on Monday, September 29, 2014
An open interval in R is of the form (a, b), where a < b +.
(1) Let M be a smooth submanifold of a smooth manifold N , and
let X, Y be smooth vector elds on M . Le
Mathematics G4402. Modern Geometry
Fall 2014
Assignment 1
Due on Monday, September 15, 2014
In this assignment:
smooth structure = C dierentiable structure
smooth map = C map
(1) (Stereographic projection) Let
S n = cfw_(x1 , . . . , xn+1 ) Rn+1 | x2 +