On an integral formula of Gauss-Bonnet-Grotemeyer

Abstract:

Let $e(p)$ and $G(p)$ be the unit outer normal and the Gauss-Kronecker curvature of an oriented closed even-dimensional hypersurface $M$ of dimension $n$ in ${E^{n + 1}}$. Then for a fixed unit vector $c$ in ${E^{n + 1}}$, we have \[ (1)\qquad \begin {array}{*{20}{c}} {\int _M^{} {{{({\text {c}}\cdot {\text {e)}}}^m}GdV = {c_{n + m}}\chi (M)/{c_{m,}}} } & {{\text {for}}\;m = 0,2,4, \cdots ,} \\ { = 0,} & {{\text {for}}\;m = 1,3,5, \cdots ,} \\ \end {array} \] where ${\text {c}} \cdot {\text {e}}$ denotes the inner product of $c$ and $e$ the area of $m$-dimensional unit sphere, and $\chi (M)$ the Euler characteristic of $M$.