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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2006 • Question 2

Question

Math 6510 • Differentiable Manifolds Winter 2006 • Question 2

Let \(\mathcal {M}_{n\times n}\) be the set of all real \(n\times n\) matrices. This set is naturally a manifold, since it can be identified with \(\mathbb {R}^{n^2}\) by choosing an ordering of entries. Let \(O(n)\subset \mathcal {M}_{n\times n}\) be the set of orthogonal matrices, i.e. matrices \(A\) with \(AA^T=I\). Prove that \(O(n)\) is a submanifold of \(\mathcal {M}_{n\times n}\). What is the dimension of \(O(n)\)?

Raw LaTeX

Let \(\mathcal{M}_{n\times n}\) be the set of all real \(n\times n\) matrices. This
set is naturally a manifold, since it can be identified with \(\mathbb{R}^{n^2}\) by
choosing an ordering of entries. Let \(O(n)\subset\mathcal{M}_{n\times n}\) be the set of
orthogonal matrices, i.e. matrices \(A\) with \(AA^T=I\). Prove that \(O(n)\) is a submanifold
of \(\mathcal{M}_{n\times n}\). What is the dimension of \(O(n)\)?

Topic Tags

Homotopy and Degree on Manifolds Lie Groups and Matrix Lie Groups Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds