NEW CURVATURE INEQUALITIES FOR HYPERSURFACES IN THE EUCLIDEAN AMBIENT SPACE

Abstract

The spread of a matrix has been introduced by Mirsky in 1956. The classical theory provides an upper bound for the spread of the shape operator in terms of the second fundamental form of a hypersurface in the Euclidean space. In the present work, we are extending our understanding of the phenomenon by proving a lower bound, inspired from an idea developed recently by X.-Q. Chang. As we are exploring the very concept of curvature on hypersurfaces, we are introducing a new curvature invariant called amalgamatic curvature and we explore its geometric meaning by proving an inequality relating it to the absolute mean curvature of the hypersurface. In our study, a new class of geometric object is obtained: the absolutely umbilical hypersurfaces.