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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2024 • Question 2

Question

Math 6510 • Differentiable Manifolds Fall 2024 • Question 2

We will identify the set of all real \(n\times n\) matrices with \(\mathbb {R}^{n^2}\) as usual. Show that \[ O(n)=\{M\mid M\text { is an orthogonal }n\times n\text { matrix}\} \] is a submanifold of \(\mathbb {R}^{n^2}\).

Raw LaTeX

We will identify the set of all real \(n\times n\) matrices with \(\mathbb{R}^{n^2}\) as usual. Show that
\[
O(n)=\{M\mid M\text{ is an orthogonal }n\times n\text{ matrix}\}
\]
is a submanifold of \(\mathbb{R}^{n^2}\).

Topic Tags

Lie Groups and Matrix Lie Groups Smooth Manifolds and Embedded Submanifolds