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On the characteristic numbers of complete manifolds of bounded curvature and finite volume. (English) Zbl 0592.53036
Differential geometry and complex analysis, Vol. dedic. H. E. Rauch, 115-154 (1985).
[For the entire collection see Zbl 0561.00010.]
The authors study non-compact Riemannian manifolds, which are complete and of finite volume, and whose sectional curvature is bounded. If M is such a manifold and \(\Omega\) the curvature of M, then P(\(\Omega)\) is a characteristic form of M for every invariant polynomial P. Thus we obtain finite numbers \(P(M,g)=\int_{M}P(\Omega)\); g the metric of M, the so- called geometric characteristic numbers. The present paper is one in a cycle of papers devoted to the study of such numbers. The main questions adressed are:
A) What values can P(M,g) assume ? B) Is P(M,g) independent of g ? C) is P(M,g) a topological invariant ?
Sample answer: The geometric Euler number is an integer, but geometric Pontrjagin numbers can be irrational. Main technical ingredients of the discussion are the possibility of exhausting M by nice compact sets and also bounds for the \(\eta\)-invariant.
Reviewer: W.Ballmann

MSC:
53C20 Global Riemannian geometry, including pinching
57R20 Characteristic classes and numbers in differential topology