Abstract

Let G be a Lie group and K a compact subgroup of G. Then the homogeneous space G/K has an invariant Riemannian metric and an invariant volume form# G . Let M and N be compact submanifolds of G/K, and I(M # gN) an "integral invariant" of the intersection M # gN . Then the integral (1) Z G I(M # gN)# G (g) is evaluated for a large class of integral invariants I. To give an informal definition of the integral invariants I considered, let X # G/K be a submanifold, h X the vector valued second fundamental form of X in G/K. Let P be an "invariant polynomial" in the components of the second fundamental form of h X . Then the integral invariants considered are of the form I P (X) = Z X P(h X )# X . If P # 1 then I P (M #gN) = Vol(M #gN ). In this case the integral (1) is evaluated for all G, K, M and N . For P of higher degree the integral (1) is evaluated when G is unimodular and G is transitive on the set on tangent spaces of each of M and N . Then, given P,...

Citations