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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2007 • Question 6

Question

Math 6510 • Differentiable Manifolds Fall 2007 • Question 6

Identify the set of all \(n\times n\) matrices with \(\mathbb {R}^{n^2}\) and let \(O(n)\) be the subset consisting of orthogonal matrices (i.e. those that satisfy \(AA^T=I\)). Show that \(O(n)\) is a submanifold of \(\mathbb {R}^{n^2}\) and compute the tangent space at \(I\).

Raw LaTeX

Identify the set of all \(n\times n\) matrices with \(\R^{n^2}\) and let \(O(n)\)
be the subset consisting of orthogonal matrices (i.e. those that satisfy
\(AA^T=I\)). Show that \(O(n)\) is a submanifold of \(\R^{n^2}\) and compute the tangent
space at \(I\).

Topic Tags

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