Isometry actions and geodesics orthogonal to submanifolds

Abstract

We obtain a condition, involving geodesics orthogonal to tangent vectors, which implies that a submanifoldmust be contained in a level set of a Lipschitz function. One application is the following theorem. Let f : Σ → M be a differentiable immersion of a connected manifold Σ in a complete noncompact manifold with nonnegative sectional curvature. Fix a ray σ in M and assume that for all point p ∈ Σ and v ∈ T _p Σ there exists a vector η orthogonal to df _p v such that the geodesic γ _η tangent to η at p is a ray asymptotic to σ. Then f(Σ) is contained in a horosphere of M associated with σ. A similar version holds in Hadamard manifolds. Another theorem studies those ideas in the context of space forms, establishing a set of equivalent conditions on a submanifold so that it is locally contained in a hypersurface invariant under the action of isometries which fix points in a given totally geodesic complete submanifold.