Total curvatures of compact manifolds immersed in euclidean space

Abstract

This paper will be concerned with some kind of total absolute curvatures of compact manifolds Xn of dimension n (without boundary) immersed in Euclidean space E n+N of dimension n+ N (N> 1). Classical Differential Geometry handled almost exclusively with «local» curvatures for such manifolds Xn (assumed sufficiently smooth) and mainly dealed with the case N = 1. The Gauss-Bonnet theorem, extended by Allendoerfer-Weil-Chern to the case n > 2, has been for years the most important, and almost the unique, result of a “global” character “. In the classical theory of convex manifolds (boundaries of convex sets) in Euclidean space, play an important role the Minkowski's «Quermassintegrale» which may be defined globally without any assumption of differentiability and also, for sufficiently smooth convex manifolds, as integrals of the symmetric functions of the principal curvatures. This classical case shows that, in order to define total curvatures of a given Xn (not necessarily convex) immersed in E n+N, one can either give directly a global definition and then try to express it as the integral of certain local curvatures, or give first a local definition (curvature at a point x= Xn) and then computing the total curvature by integrating this local curvature over Xn. The last method makes necessary some assumptions of smoothness for Xn. A noteworthy example of such curvatures is those introduced by H. Weyl in a classical paper on the volume of tubes. These Weyl's curvatures have been used by Chern to get a general kinematic formula in integral geometry for compact sub manifolds of E n+N . For more general subsets of E n+N an analogous formula was given by H. Pederer whose «curvature measures » are an extension of the Weyl's curvatures