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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2023 • Question 5

Question

Math 6510 • Differentiable Manifolds Fall 2023 • Question 5

Identify \(\mathcal {M}(2)\), the set of two-by-two matrices, with \(\mathbb {R}^4\). Let \(SL(2,\mathbb {R})\subset \mathcal {M}(2)\) be the set of matrices with determinant one. Show that \(SL(2,\mathbb {R})\) is a smooth submanifold and calculate the tangent space \(T_{\mathrm {id}}SL(2,\mathbb {R})\) as a subspace of \(T_{\mathrm {id}}\mathcal {M}(2)=\mathbb {R}^4\).

Raw LaTeX

Identify \(\mathcal{M}(2)\), the set of two-by-two matrices, with \(\mathbb{R}^4\). Let \(SL(2,\mathbb{R})\subset\mathcal{M}(2)\) be the
set of matrices with determinant one. Show that \(SL(2,\mathbb{R})\) is a smooth submanifold and calculate
the tangent space \(T_{\mathrm{id}}SL(2,\mathbb{R})\) as a subspace of \(T_{\mathrm{id}}\mathcal{M}(2)=\mathbb{R}^4\).

Topic Tags

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