Kinematische Berührformeln in Riemannschen homogenen Räumen. (Kinematic contact formulas in Riemannian homogeneous spaces) (Q1826151)

For two compact smooth submanifolds \(M_ 1\), \(M_ 2\) of a space of constant curvature, let A be the set of all rigid motions g for which \(gM_ 2\) intersects \(M_ 1\). Classical formulae of integral geometry express integrals over A, with respect to the invariant measure, of certain geometric functions of the intersection \(M_ 1\cap gM_ 2\) in terms of curvature invariants of \(M_ 1\) and \(M_ 2\). If intersection is replaced by touching, suitably defined, the corresponding sets of rigid motions are of lower dimension. One can still define natural measures for sets of touching positions and express them in terms of curvature invariants. In two former papers [Kinematic contact formulas in spaces of constant curvature I and II, Abh. Math. Semin. Univ. Hamb. (to appear) (for the preview of part I, cf. Zbl 0644.53064)] the author has investigated various touching situations and obtained integral-geometric formulae for these. In the present paper, he develops a more general theory in Riemannian homogeneous spaces. The measures for the sets of touching positions are derived from a suitably defined metric. In the general case, the touching situations themselves have to satisfy restrictive assumptions in order that (more or less) explicit results can be obtained.