Abstract
Let G be a Lie group and H its subgroup, and $M^{q}$ , $N^{r}$ two submanifolds of dimensions q, r, respectively, in the Riemannian homogeneous space $G/H$ . A kinematic integral formula for the angle between the two intersected submanifolds is obtained.
Highlights
Kinematic formulas in integral geometry are important and useful
In [ ], he provided integral formulas for the quantities introduced by Weyl for the volume of tubes
These formulas complement the fundamental kinematic formula, which only deals with hypersurfaces
Summary
Kinematic formulas in integral geometry are important and useful. At the beginning of [ ], Chern said: ‘one of the basic problems in integral geometry is to find explicit formulas for the integrals of geometric quantities over the kinematic density in terms of known integral invariants’. Let I(M ∩ gN) = χ(M ∩ gN) be the Euler characteristic of M ∩ N of domains M and N in Rn, G χ (M ∩ gN) dg can be expressed explicitly by the integrals of elementary symmetric functions of principal curvatures over the boundaries and the Euler characteristics of M, N This well-known kinematic formula in integral geometry is due to Chern [ , ]. Assume that I(Mq ∩ gNr) = μ(Mq ∩ gNr) is one of the integral invariants from the Weyl tube formula, ( ) leads to the ChernFederer kinematic formula for submanifolds of Rn [ ]. Let Mq and Nr be two submanifolds in Rn, Mq fixed, and gNr moving under the rigid motion g of Rn with the kinematic density dg.
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