zbMATH — the first resource for mathematics

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

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