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e2_m7350_s2011_v1_0

# e2_m7350_s2011_v1_0 - Exam 2 Math 7350 Spring 2011 Problem...

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Exam 2: Math 7350 Spring 2011 Problem 1. (10) Let V be a real vector space of dimension N . Prove that dim ( Σ k ( V ) ) = ± N + k - 1 k ² = ( N + k - 1)! k !( N - 1)! . Problem 2. (10) Let ( M,g 1 ) and ( N,g 2 ) be Riemannian manifolds. Suppose F : M N is a smooth map such that F * ( g 2 ) = g 1 . Prove that F is an immersion. Problem 3. (10) Let N > 3 be an integer and let V be a real vector space of dimension N . Prove that Σ 3 ( V ) Λ 3 ( V ) 6 = T 3 ( V ). Problem 4. (10) Suppose M and N are oriented smooth manifolds and F : M N is a local diﬀeomorphism. If M is connected, prove that F is either orientation-preserving or orientation-reversing. Problem 5. (10) Let ω be the ( N - 1)-form on R N \ { 0 } deﬁned by ω = k x k - N N X i =1 ( - 1) i - 1 x i d x 1 ∧ ··· ∧ d d x i ∧ ··· ∧ d x N , where ‘hat’ denotes deletion of a term. Prove that ω is closed but not exact on R N \ { 0 } . Problem 6. (10)
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