Index formulas for geometric Dirac operators in Riemannian foliations

With regards to certain Riemannian foliations we consider Kasparov pairings of leafwise and transverse Dirac operators. Relative to a pairing with a transversal class we commence by establishing an index formula for foliations with leaves of nonpositive sectional curvature. The underlying ideas are then developed in a more general setting leading to pairings of images under the Baum-Connes map in geometricK-theory with transversal classes. Several ideas implicit in the work of Connes and Hilsum-Skandalis are formulated in the context of Riemannian foliations. From these we establish the notion of a dual pairing inK-homology and a theorem of the Grothendieck-Riemann-Roch type.R. G. D. was supported by The National Science Foundation under Grant No. DMS-9304283.J. F. G. and F. W. K. were supported in part by The National Science Foundation under Grant No. DMS-9208182.F. W. K. was also supported in part by an Arnold O. Beckman Research Award from the Research Board of the University of Illinois.