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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2013 • Question 6

Question

Math 6510 • Differentiable Manifolds Fall 2013 • Question 6

Regard \(\SO (n)\) as a subset of the space of \(n\times n\) matrices, which in turn is identified with \(\mathbb {R}^{n^2}\). Show that \(\SO (n)\) is a submanifold and identify the tangent space to \(\SO (n)\) at the identity \(I\) (i.e. the Lie algebra of \(\SO (n)\)).

Raw LaTeX

Regard \(\SO(n)\) as a subset of the space of \(n\times n\) matrices, which in
turn is identified with \(\R^{n^2}\). Show that \(\SO(n)\) is a submanifold and
identify the tangent space to \(\SO(n)\) at the identity \(I\) (i.e. the Lie algebra of
\(\SO(n)\)).

Topic Tags

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