An extension of Rauch comparison theorem to glued Riemannian spaces

Abstract

A glued Riemannian space is obtained from Riemannian manifolds $M_1$ and $M_2$ by identifying their isometric submanifolds $B_{1}$ and $B_{2}$. A curve on a glued Riemannian space which is a geodesic on each Riemannian manifold and satisfies certain passage law on the identified submanifold $B \mathrel{:=} B_{1} \cong B_{2}$ 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 Rauch's theorem which gives a comparison of the lengths of Jacobi fields along geodesics in different Riemannian manifolds to $B$-Jacobi fields along $B$-geodesics in different glued Riemannian spaces.