We show that the geodesic flow and the exponential map of a $C^k$ submanifold of $\mathbb{R}^n$ with $k\geq 2$ are of class $C^{k-1}$.
Killing tensor fields on the 2-torus
Surfaces of section for geodesic flows of closed surfaces
Differential invariants for cubic integrals of geodesic flows on surfaces
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