𝐺-deformations and some generalizations of H. Weyl’s tube theorem

Abstract

We prove an invariance theorem for the volumes of tubes about submanifolds in arbitrary analytic Riemannian manifolds under G G -deformations of the second order. For locally symmetric spaces or two-point homogeneous spaces we give stronger invariance theorems using only G G -deformations of the first order. All these results can be viewed as generalizations of the result of H. Weyl about isometric deformations and the volumes of tubes in spaces of constant curvature. They are derived from a new formula for the volume of a tube about a submanifold.