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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2018 • Question 4

Question

Math 6510 • Differentiable Manifolds Spring 2018 • Question 4

Let \(G\) be a Lie group with Lie algebra \(\mathfrak {g}\) and exponential map \(\exp :\mathfrak {g}\to G\). For a nonzero \(v\in \mathfrak {g}\), let \(L\) be the line spanned by \(v\). Prove that \(\exp |_L:L\to G\) is an immersion.

Raw LaTeX

Let \(G\) be a Lie group with Lie algebra \(\mathfrak{g}\) and exponential map
\(\exp:\mathfrak{g}\to G\). For a nonzero \(v\in\mathfrak{g}\), let \(L\) be the line
spanned by \(v\). Prove that \(\exp|_L:L\to G\) is an immersion.

Topic Tags

Immersions, Submersions, and Embeddings Lie Algebras and Exponential Maps Lie Groups and Matrix Lie Groups