Integral formulae on hypersurfaces in Riemannian manifolds

Abstract

The coefficients of the complete set of n fundamental forms of a hypersuface Vn−1 imbedded in an n-dimensional Riemannian space Vn, as recently introduced[(5)], are used to construct certain tensor fields over Vn−1 which display some remarkable features. In particular, the divergences of these tensor fields can be expressed very simply in terms of polynomials involving the curvature tensor of Vn, the coefficients of the n fundamental forms, and the rth curvatures of Vn−1. As the result of an application of the generalized divergence theorem of Gauss to these relations a set of integral formulae on Vn−1 is obtained. The integrands of these integral formulae can be expressed very simply in terms of the n fundamental forms of Vn−1. By successive specialization it is indicated how known integral theorems([2], [3], [6], [7], [8]) can be derived as particular cases, which is possible partly as a result of the fact that the polynomial referred to above vanishes identically whenever Vn is a space of constant curvature.