A Morse index theorem for geodesics on a glued Riemannian space

Abstract

A glued Riemannian space is obtained from Riemannian manifolds M₁ and M₂ by identifying their isometric submanifolds B₁ and B₂. A curve on a glued Riemannian space which is a geodesic on each Riemannian manifold and satisfies certain passage law on the identified submanifold B := B₁ ≅ B₂ is called a B-geodesic. Considering the variational problem with respect to arclength L of piecewise smooth curves through B, a critical point of L is a B-geodesic. A B-Jacobi field is a Jacobi field on each Riemannian manifold and satisfies certain passage condition on B. In this paper, we extend the Morse index theorem for geodesics in Riemannian manifolds to the case of a glued Riemannian space.