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01 - manifolds supplement

# 01 - manifolds supplement - INTRODUCTION TO MANIFOLDS I...

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INTRODUCTION TO MANIFOLDS — I Supplementary problems Matrix manifolds Problem 1. Let M = Mat n × n R n 2 be the set of all square matrices. Prove that the map Ad C : A 7→ C - 1 AC, det C 6 = 0 , defines a diffeomorphism of the manifold M onto itself. Problem 2. The same question about the manifold N M of matrices of determinant 1. Problem 3. The same question but for the map M 3 A 7→ B - 1 AC, det B, det C 6 = 0 . (1) Problem 4. Prove that for any two matrices of the same rank there exists a diffeomorphism M M of the form (1) taking one into the other. Problem 5. Prove that det( E + εB ) = 1 + ε tr B + O ( ε 2 ) . Using the previous problem, deduce the formula for the first order term in the expansion det( A + εB ), when det A 6 = 0. Problem 6. Prove that the set of matrices M r M of the rank r 6 n is a smooth submanifold in M . Is this submanifold closed? Compute its dimension. (Answer: n 2 - ( n - r ) 2 = 2 nr - r 2 .) Partition of unity Everywhere below M stands for a smooth n -dimensional manifold.

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01 - manifolds supplement - INTRODUCTION TO MANIFOLDS I...

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