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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2014 • Question 1

Question

Math 6510 • Differentiable Manifolds Fall 2014 • Question 1

Identify \(\mathcal {M}(2)\), the set of two-by-two matrices, with \(\mathbb {R}^4\). Let \(\mathrm {SL}(2,\mathbb {R})\subset \mathcal {M}(2)\) be the set of matrices with determinant one. Show that \(\mathrm {SL}(2,\mathbb {R})\) is a smooth submanifold and calculate the tangent space \(T_{id}\mathrm {SL}(2,\mathbb {R})\) as a subspace of \(T_{id}\mathcal {M}(2)=\mathbb {R}^4\). Is the line \[ \begin {pmatrix}t&0\\0&1\end {pmatrix},\quad t\in \mathbb {R} \] transverse to \(\mathrm {SL}(2,\mathbb {R})\)? Justify your answer.

Raw LaTeX

Identify \(\mathcal{M}(2)\), the set of two-by-two matrices, with \(\R^4\). Let
\(\SL(2,\R)\subset\mathcal{M}(2)\) be the set of matrices with determinant one. Show
that \(\SL(2,\R)\) is a smooth submanifold and calculate the tangent space
\(T_{id}\SL(2,\R)\) as a subspace of \(T_{id}\mathcal{M}(2)=\R^4\). Is the line
\[
  \begin{pmatrix}t&0\\0&1\end{pmatrix},\quad t\in\R
\]
transverse to \(\SL(2,\R)\)? Justify your answer.

Topic Tags

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