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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2001 • Question 2

Question

Math 6510 • Differentiable Manifolds Fall 2001 • Question 2

Recall that an \(n\times n\) real matrix \(M\) is orthogonal if \[ \langle Mv,Mw\rangle =\langle v,w\rangle \] for all \(v,w\in \mathbb {R}^n\) (and \(\langle \cdot ,\cdot \rangle \) is the inner product). Let \(O(n)\) denote the set of all orthogonal \(n\times n\) matrices, viewed as a subset of \(\mathbb {R}^{n^2}\). Show that \(O(n)\) is a manifold, and compute its dimension.

Raw LaTeX

Recall that an \(n\times n\) real matrix \(M\) is orthogonal if
\[
  \langle Mv,Mw\rangle=\langle v,w\rangle
\]
for all \(v,w\in\mathbb{R}^n\) (and \(\langle\cdot,\cdot\rangle\) is the inner product).
Let \(O(n)\) denote the set of all orthogonal \(n\times n\) matrices, viewed as a subset
of \(\mathbb{R}^{n^2}\). Show that \(O(n)\) is a manifold, and compute its dimension.

Topic Tags

Homotopy and Degree on Manifolds Lie Groups and Matrix Lie Groups Pullbacks and Wedge Products Smooth Manifolds and Embedded Submanifolds