On the Gauss-Bonnet theorem for complete manifolds
Transactions of the American Mathematical Society
For a manifold diffeomorphic to the interior of a compact manifold with boundary, several classes of complete metrics are given for which the Gauss-Bonnet Theorem is valid. Introduction. For a compact oriented Riemannian manifold M, the Gauss-Bonnet Theorem states that x(M) = fME(g), where E(g) is the Euler form for the metric g. For noncompact manifolds the theorem is known to hold on finitely connected, finite volume Riemann surfaces and for the invariant metric on quotients of symmetric
... s of symmetric spaces by discrete, torsion-free, arithmetic subgroups. The proofs are due to [Hu] and Harder [Ha] respectively (see also [Gr, p. 84; CG]). A manifold in either of these classes is diffeomorphic to the interior of a compact manifold with boundary. This is classical for Riemann surfaces, while for arithmetic domains the result follows from the reduction theory of Borel and Harish-Chandra. Elementary Morse theory thus guarantees that there are coordinates on a collar dM x R+ of the boundary in which the metric may be written g = gi(x, i) + dt2, where x G dM and t G R+. In [Ro] this form of the metric gives rise to a short proof of Huber's result. In §1 we exhibit, through the use of moving frames, several classes of complete metrics for which Gauss-Bonnet is valid. For example, in Theorem 1.9 we show that if g is a warped product metric, g = f2(t)gi + dt2 for gi a metric on dM, then for Gauss-Bonnet it suffices that / -> 0 and /' -> 0 as t -> oo. Since E(g) depends on the second derivatives of g, it is surprising at first glance that only first order conditions on / are needed. However, roughly speaking, the warped product structure controls the second order information. All the metrics in §1 are more or less modeled on (1.14), Borel's explicit form of the invariant metric for arithmetic quotients of split rank-one symmetric spaces. Even with this explicit form of the metric, Harder's result is not trivial (see Theorem 1.13). The calculation gives a glimpse of the delicate interplay between the algebra and geometry of these spaces. In §2 we give some other metrics for which Gauss-Bonnet holds. In particular, we show how to construct many surfaces of positive curvature and infinite volume with Gauss-Bonnet. This contrasts with Huber's result, since in some sense finite volume surfaces must be mostly negatively curved near infinity.