New curvature inequalities for hypersurfaces in the euclidean ambient space

摘要

The spread of a matrix is introduced by Mirsky in 1956 in [20]. 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 study the concept of curvature on hypersurfaces, we introduce a new curvature invariant called amalgamatic curvature and explore its geometric meaning by proving an inequality related to the absolute mean curvature of the hypersurface. In our study, a new class of geometric objects is obtained: the absolutely umbilical hypersurfaces.