An Integral Formula and Inverse Fundamental Forms on Hypersurfaces In Riemannian Manifolds

Abstract

In this study, the coefficients of the p-fundamental forms of a hypersurface N imbedded in n-dimensional Riemannian space M were expressed in terms of the coefficients of first and se- cond fundamental forms. Then, by means of Cayley-Hamilton theorem, the inverse S-1 of the shape operatör S on the hypersurface N was vvritten as the combinations of the powers of S and the curvatures K n ... K p Thus the new fundamental forms and some properties of them cal- led the inverse fundamental forms, were defined and investigated. As a result of an application of the generalized divergence theorem of Gauss to the divergence relations of certain tensor fi- elds över the region R of N that can be expressed in terms of polynomials involving the new de­ fined curvatures of M an integral formula was obtained.