English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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 So n gS, of two C2-smooth hypersurfaces S0, S1, i.e., fG(fs0ngs, H2kda) 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 R2n , i.e., sufficient conditions for one domain Kg to contain, or to be contained in, another domain KaW . 0. INTRODUCTION Let MPI, Nq be submanifolds of dimensions p, q, respectively, in a homogeneous space G/H and let I be an integral invariant (e.g., volume, surface area, etc.) of the submanifold MP n gNq . Then many works in integral geometry have been concerned with computing integrals of the following type (0. 1) I(MP n gNq) dg, {gEG: MpngNq70} where dg is the normalized kinematic density of G. For example in the case that G is the group of motions in an n-dimensional Euclidean space Rn, MP and Nq are submanifolds of Rn and I(MP n gNq) = Vol(MP n gNq) evaluation of (0.1) leads to the formulas of Poincare, Blaschke, Santal6 and others (see [1]). R. Howard [10] obtained a kinematic formula for I(MP n gNq) = Vol(MP n gNq) in a homogeneous space. If I(M n gN) = X(M n gN), where X%() is the Euler-Poincare characteristic of the intersection M n gN of domains M and N in Rn with smooth boundaries, then (0.1) leads to S. S. Chern's kinematic fundamental formula [2]. Next, assume that I(MP n gNq) = #u(MP n gNq) is one of the integral invariants from the Weyl tube formula. Then (0.1) leads to the Chern-Federer kinematic formula [22] Received by the editors March 9, 1993 and, in revised form, September 15, 1993. 1991 Mathematics Subject Classification. Primary 52A22, 53C05; Secondary 5 lM1 6.

Kinematic formulas for mean curvature powers of hypersurfaces and Hadwiger’s theorem in 𝑅²ⁿ