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.


53D25 Geodesic flows in symplectic geometry and contact geometry
53C22 Geodesics in global differential geometry
53C20 Global Riemannian geometry, including pinching
Full Text: DOI arXiv Euclid