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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1994 • Question 19

Question

Math 6510 • Differentiable Manifolds Fall 1994 • Question 19

Let \(M^2\) be a smooth closed connected orientable surface with Gauss curvature \(K(x)<0\) and genus \(g\). Let \(\{\gamma _1,\ldots ,\gamma _\ell \}\) be a system of pairwise nonintersecting closed geodesics of \(M\). (A closed geodesic is a smoothly embedded loop \(\gamma _i:S^1\to M\) whose geodesic curvature vanishes identically \(\kappa _i=0\).) Show that \(\ell \le 3g-3\). (Hint: can two nonintersecting closed geodesics be freely homotopic?)

Raw LaTeX

Let \(M^2\) be a smooth closed connected orientable surface with Gauss
curvature \(K(x)<0\) and genus \(g\). Let \(\{\gamma_1,\ldots,\gamma_\ell\}\) be a system
of pairwise nonintersecting closed geodesics of \(M\). (A closed geodesic is a smoothly
embedded loop \(\gamma_i:S^1\to M\) whose geodesic curvature vanishes identically
\(\kappa_i=0\).) Show that \(\ell\le 3g-3\). (Hint: can two nonintersecting closed
geodesics be freely homotopic?)

Topic Tags

Homotopy Equivalence, Retracts, and Contractibility Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups