26.
The coefficients of the complete set of n fundamental forms of a hypersuface V
n−1 imbedded in an n-dimensional Riemannian space V
n, as recently introduced
[(5)], are used to construct certain tensor fields over V
n−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 V
n, the coefficients of the n fundamental forms, and the r
th curvatures of V
n−1. As the result of an application of the generalized divergence theorem of Gauss to these relations a set of integral formulae
on V
n−1 is obtained. The integrands of these integral formulae can be expressed very simply in terms of the n fundamental forms of
V
n−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 V
n is a space of constant curvature.
This research was supported by the National Research Council of Canada.
Entrata in Redazione il 21 agosto 1970.
相似文献