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Closing geodesics in \(C^1\) topology. (English) Zbl 1258.53087

Let \((M,g)\) be a smooth compact manifold of dimension \(n\geq2\). If \(g\) is a \(C^k\) metric on \(M\) for \(k\geq3\), let \(|v|_x^g\) be the norm of a tangent vector \(v\); the case \(k=\infty\) is permitted. Choose \(x\in M\) and a unit tangent direction \(v\in T_xM\). Let \(\epsilon>0\) be given. The author shows that there exists a conformally equivalent metric \(\tilde g=e^fg\), where \(f\) is of class \(C^{k-1}\), so that \(\|f\|_{C^1}<\epsilon\) and there exists \((\bar x,\bar v)\), with \(d((x,v),(\bar x,\bar v))<\epsilon\), such that the \(\bar g\)-geodesic starting at \(\bar x\) in the direction \(\bar v\) is periodic.

MSC:

53D25 Geodesic flows in symplectic geometry and contact geometry
53C22 Geodesics in global differential geometry
53C20 Global Riemannian geometry, including pinching
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