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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1997 • Question 14

Question

Math 6510 • Differentiable Manifolds Fall 1997 • Question 14

Consider the set of matrices \[ O(n)=\{A\in \mathcal {M}_n:AA^T=1\} \] where \(\mathcal {M}_n\cong \mathbb {R}^{n^2}\) is the collection of all \(n\times n\) matrices and \(A^T\) means the transposition of \(A\). Show that \(O(n)\) is a smooth submanifold of \(\mathcal {M}_n\). Describe the set of matrices which form the tangent space to \(O(n)\) at the identity matrix.

Raw LaTeX

Consider the set of matrices
\[
  O(n)=\{A\in\mathcal{M}_n:AA^T=1\}
\]
where \(\mathcal{M}_n\cong\mathbb{R}^{n^2}\) is the collection of all \(n\times n\)
matrices and \(A^T\) means the transposition of \(A\). Show that \(O(n)\) is a smooth
submanifold of \(\mathcal{M}_n\). Describe the set of matrices which form the tangent
space to \(O(n)\) at the identity matrix.

Topic Tags

Lie Groups and Matrix Lie Groups Smooth Manifolds and Embedded Submanifolds Tangent Spaces and Differential Maps