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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2023 • Question 1

Question

Math 6510 • Differentiable Manifolds Fall 2023 • Question 1

Let \(\alpha :G\times M\to M\) be an action of a Lie group \(G\) on a \(C^\infty \) manifold \(M\) such that \(\alpha \) is a continuous function. Prove or find a counterexample: if \(\alpha |_{G\times \{x\}}:G\to M\) is a \(C^\infty \) immersion for every \(x\in M\), then the function \(\alpha \) is \(C^\infty \).

Raw LaTeX

Let \(\alpha:G\times M\to M\) be an action of a Lie group \(G\) on a \(C^\infty\) manifold \(M\) such that \(\alpha\)
is a continuous function. Prove or find a counterexample: if \(\alpha|_{G\times\{x\}}:G\to M\) is a \(C^\infty\) immersion
for every \(x\in M\), then the function \(\alpha\) is \(C^\infty\).

Topic Tags

Homotopy and Degree on Manifolds Immersions, Submersions, and Embeddings Lie Group Actions and Homomorphisms Lie Groups and Matrix Lie Groups