A Dehn type quantity for Riemannian manifolds
Abstract
We look at the functional Y(M) = intM K(x) dV(x) for compact Riemannian 2d-manifolds M, where K(x) = (2d)! (d!)-1 (4pi)-d intT prodk=1d Kt2k,t2k+1(x) dt involves products of d sectional curvatures Kij(x) averaged over the space T sim O(2d) of all orthonormal frames t=(t1, ... ,t2d). A discrete version Ydisc(M) with Kd(x) = (d!)-1 (4pi)-d sumsigma prodk=1d Ksigma(2k-1),sigma(2k) sums over all permutations sigma of 1,..,2d. Unlike Euler characteristic which by Gauss-Bonnet-Chern is intM KGBC dV=X(M), the quantities Y or Ydisc are in general metric dependent. We are interested in Y(M)-X(M) because if M has curvature sign e, then Y(M) ed and Ydisc(M) are positive while X(M) ed>0 is only conjectured. We compute Ydisc in a few concrete examples like 2d-spheres, the 4-manifold CP2, the 6 manifold SO(4) or the 8-manifold SU(3).