Kinematic formulas for mean curvature powers of hypersurfaces and Hadwiger’s theorem in $\textbf {R}^ {2n}$

Abstract:

We first discuss the theory of hypersurfaces and submanifolds in the m-dimensional Euclidean space leading up to high dimensional analogues of the classical Euler’s and Meusnier’s theorems. Then we deduce the kinematic formulas for powers of mean curvature of the $(m - 2)$-dimensional intersection submanifold ${S_0} \cap g{S_1}$ of two ${C^2}$-smooth hypersurfaces ${S_0}$, ${S_1}$, i.e., ${\smallint _G}({\smallint _{{S_0} \cap g{S_1}}}{H^{2k}}d\sigma )dg$. Many well-known results, for example, the C-S. Chen kinematic formula and Crofton type formulas are easy consequences of our kinematic formulas. As direct applications of our formulas, we obtain analogues of Hadwiger’s theorem in ${\mathbb {R}^{2n}}$, i.e., sufficient conditions for one domain ${K_\beta }$ to contain, or to be contained in, another domain ${K_\alpha }$.