Isometry actions and geodesics orthogonal to submanifolds
  We investigate submanifolds in space forms such that every geodesic orthogonal to the submanifold intersects a fixed totally geodesic submanifold. We obtain an application to horospheres in Hadamard manifolds. 
Antonio J. Di Scala, S\'ergio Mendon\c{c}a, Heudson Mirandola and   Gabriel Ruiz-Hern\'andez
math.DG