Curvature measures on manifolds: minimal immersions

Abstract

The purpose of this work is to give some results on the various curvature measures on manifolds and also have a brief look at minimal immersions of manifolds in Riemannian spaces. With regard to the former, the first chapter deals with the 1th TAC as defined by Chen [9]. In Chapter II we look at minimal immersions of compact manifolds in Riemannian spaces and in particular at pseudo-umbilical immersions - the term first introduced by Otsuki. The two more familiar curvatures are the scalar curvature, and the mean curvature, and in Chapter III we define the ɑ(^th) scalar curvature. Finally we look at submanifolds with constant mean curvature. Lastly, in Chapter IV, a differential equation is derived for "stable hypersurfaces". A hypersurface is said to be 'stable' if б│(_M(^n)) <Ḫ,Ḫ>(^n/2) * 1 = 0 for any normal variation of the integral. A particular case of this problem, (i.e. for surfaces in E(^3)) was first considered by Hombu. A bibliography follows Chapter IV.