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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1996 • Question 9

Question

Math 6510 • Differentiable Manifolds Fall 1996 • Question 9

Let \(S\) be a surface with Riemannian metric whose curvature \(K(p)\) is nonpositive at all points \(p\) of \(S\). Let \(\gamma :\mathbb {R}\to S\) be a geodesic such that \(\gamma (t)=\gamma (t+1)\) for all \(t\) and \(\gamma \) represents the trivial element of the fundamental group \(\pi _1(S,\gamma (0))\). Prove1 1Without referring to Cartan-Hadamard theorem. that the mapping \(\gamma \) is constant.

Raw LaTeX

Let \(S\) be a surface with Riemannian metric whose curvature \(K(p)\) is
nonpositive at all points \(p\) of \(S\). Let \(\gamma:\mathbb{R}\to S\) be a geodesic such
that \(\gamma(t)=\gamma(t+1)\) for all \(t\) and \(\gamma\) represents the trivial element
of the fundamental group \(\pi_1(S,\gamma(0))\). Prove\footnote{Without referring
to Cartan-Hadamard theorem.} that the mapping \(\gamma\) is constant.

Topic Tags

Fundamental Groups Homotopy Groups Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups