By Peter L Duren; Richard Askey; Uta C Merzbach; Harold M Edwards

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28 Let (Mn,g) be open, completeandoffinitetopological type. Assume that in an open neighbourhood of each end c, TMlc = W 2 EB ... EB Wr< EB IR and the metric is of the form fi(r)g2 EB ... EB l;Jr)gr< + dr2 with gi a metric on Wi' If fi (r) ----+ 0, ff( r) ----+ 0 and fj fi- I and (fj fi- I )' are bounded r---+oo r-too for all r, i, j then X(M) = J E(g). D Example. Let M\G/ K be an arithmetic quotient of an evendimensional split rank-one symmetric space. Then at each component 8Mi of 8M, 8M is the total space of a fibration over a torus TI with a torus T2 as fiber.

EB Wr< EB IR and the metric is of the form fi(r)g2 EB ... EB l;Jr)gr< + dr2 with gi a metric on Wi' If fi (r) ----+ 0, ff( r) ----+ 0 and fj fi- I and (fj fi- I )' are bounded r---+oo r-too for all r, i, j then X(M) = J E(g). D Example. Let M\G/ K be an arithmetic quotient of an evendimensional split rank-one symmetric space. Then at each component 8Mi of 8M, 8M is the total space of a fibration over a torus TI with a torus T2 as fiber. We have T MlvxlR = WI EB W 2 EB IR for open V c 8M where the fibration restricted to V is trivial.

Ik and fMLk = f'~Lk' If additionally P = 1 and (M, g) is complete then the corresponding characteristic numbers coincide. 14. 51 It it possible that b,lC~,l(Bo, 1, f) = 0. Let (M2, g) be an infinitely connected open complete Riemannian manifold with bounded sectional curvature K, K = K+ - K_. K, K 2 0 { - K, K:==; 0 K+ = { 0, K < 0 ,K_ = 0, K > 0 ,Then there holds J K_ dvol = 00 (cf. [43], theorem 13). In particular J IKI dvol = 00 which implies J I[2W(g)1 dvol = 00. The proof essentially relies on the GauB-Bonnet theorem (as one would expect) for compact surfaces.

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