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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1995 • Question 11

Question

Math 6510 • Differentiable Manifolds Fall 1995 • Question 11

Let \(M_2(\mathbb {R})\) be the set of all \(2\times 2\) matrices with real coefficients, and identify it with \(\mathbb {R}^4\) in the usual way. Let \(SL(2,\mathbb {R})\) be the subset of \(M_2(\mathbb {R})\) consisting of those matrices with determinant equal to one.

(a) Show \(SL(2,\mathbb {R})\) is a submanifold of \(M_2(\mathbb {R})\).

(b) Exhibit the tangent space to \(SL(2,\mathbb {R})\) at the identity matrix.

Raw LaTeX

Let \(M_2(\mathbb{R})\) be the set of all \(2\times 2\) matrices with real
coefficients, and identify it with \(\mathbb{R}^4\) in the usual way. Let
\(SL(2,\mathbb{R})\) be the subset of \(M_2(\mathbb{R})\) consisting of those matrices
with determinant equal to one.

\textbf{(a)} Show \(SL(2,\mathbb{R})\) is a submanifold of \(M_2(\mathbb{R})\).

\textbf{(b)} Exhibit the tangent space to \(SL(2,\mathbb{R})\) at the identity matrix.

Topic Tags

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