Isometry actions and geodesics orthogonal to submanifolds

We obtain a condition, involving geodesics orthogonal to tangent vectors, which implies that a submanifold must be contained in a level set of a Lipschitz function. One application is the following theorem. Let f : S → M be a differentiable immersion of a connected manifold S in a complete noncompact manifold with nonnegative sectional curvature. Fix a ray σ in M and assume that for all point p ∈ S and v ∈ TS_p there exists a vector η orthogonal to df S_p such that the geodesic γ tangent to η at p is a ray asymptotic to σ. Then f(S) 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.