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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2000 • Question 1

Question

Math 6510 • Differentiable Manifolds Fall 2000 • Question 1

Let \(\Sigma \) be a smooth, compact, orientable surface (2 dimensional manifold) of genus \(g\). Prove that \(\Sigma \) admits a smooth non-vanishing vectorfield if and only if \(g=1\).

Raw LaTeX

Let \(\Sigma\) be a smooth, compact, orientable surface (2 dimensional
manifold) of genus \(g\). Prove that \(\Sigma\) admits a smooth non-vanishing vectorfield
if and only if \(g=1\).

Topic Tags

Homotopy and Degree on Manifolds Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups