Isometry actions and geodesics orthogonal to submanifolds (Q2343144)

The authors provide a condition, involving geodesics orthogonal to vectors tangent to a given submanifold \(\Sigma\), which implies that \(\Sigma\) is contained in a level set of a Lipschitz function. More precisely, they prove the following Theorem: Let \(f:\Sigma\to M\) be a smooth immersion of a connected manifold \(\Sigma\) into a Riemannian manifold \(M\). Let \(G:M\to\mathbb{R}\) be a Lipschitz function with a Lipschitz constant \(C\). Assume that for any \(p\in\Sigma\) and \(v\in T_p\Sigma\) there exists a non-zero vector \(\eta\in T_{f(p)}M\) orthogonal to \(df_p(v)\) such that the geodesic \(\gamma_\eta: [0, 1]\to M\) satisfies \(|G(f(p)) - G(\gamma_\eta(1))| = C\cdot L(\gamma_\eta)\). Then \(f(\Sigma)\) is contained in a level set of \(G\). Several corollaries (for Hadamard manifolds, spaces of constant curvature and so on) are derived.