The volume form. 1. Orientation on vector spaces and manifolds. (1) Let V be a vector space of dimension n 1. Recall that an orientation on V is specied by choosing a particular basis e1 , . . . , en of V . If v1 , . . . , vn is any other basis of V , thi
H/wk 10. Due Friday, April 13. NAME: 1. Let be an r-form in a manifold M n such that for some r-chain in M with = 0 we have = 0. Prove that is not exact. Solution. Suppose that is exact, so that = d for some (r 1)-form . Then by Stokes Theorem =
d =
=
0
Riemannian Connections 1. General Denition of a Riemannian Connection. Let (M, g ) be a smooth manifold with a smooth Riemannian metric g . Let V be the set of all smooth vector elds on M . A connection or a covariant derivative on M is an operator : V V
Math 481 Exam 2 (SOLUTIONS), Friday, April 6, 2007 1. Let W be a (0, 2)-tensor on a 2-manifold M such that in a chart (U, = (x1 , x2 ) we have W 1,1 = x1 + 5, W i,j = 0 for (i, j ) = (1, 1). y 2 = x1 . Let (V, = (y 1 , y 2 ) be another chart such that V =
H/wk 9 (SOLUTIONS). Due Wednesday, April 4. 1. For n r 1 dene the following function I : r (Rn ) r1 (Rn ) from the space of smooth r-forms on Rn to the space of smooth (r 1)-forms of Rn . For =
i1 <i2 <ir r 1
i1 .ir dxi1 dxir
I ( ) =
i1 <i2 <ir j =1 0
(1)
Math 481 H/wk 12 (Solutions), Due Friday, April 27 1. Let X = (3x z, yx2 , xyz + 1) and Y = (exz , y + x, z ). (a) Compute DX Y . (b) For c(t) = (t, t2 , 1 3t) compute Dc Y . Solution. (a) We have: DX Y = (3x z )De1 Y + yx2 De2 Y + (xyz + 1)De3 Y = (3x z