Closest point of the cut locus to submanifold (Q1118832)

Let M be a complete Riemannian manifold, L a compact connected submanifold of M and \(c_ 1,c_ 2: [0,\infty]\to M\) two unit speed geodesics of M emanating perpendicularly from L. Moreover, suppose that there exists an \(r>0\) such that \(x:=c_ 1(r)=c_ 2(r)\), but \(\dot c{}_ 1(r)\neq \dot c_ 2(r)\), where r is the distance from x to L; in particular, x is a cut point of L, but x is assumed not to be a focal point. The main result states that then in every neighbourhood of x there are cut points of L, which are nearer to L than x. For the proof the author derives a generalization of the Gauß-Lemma describing the relation of the geometry of the normal bundle of L and the geometry of M (via the normal exponential map).