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**A vanishing theorem for piecewise constant curvature spaces.**
*(English)*
Zbl 0612.53024

Curvature and topology of Riemannian manifolds, Proc. 17th Int. Taniguchi Symp., Katata/Jap. 1985, Lect. Notes Math. 1201, 33-40 (1986).

[For the entire collection see Zbl 0583.00022.]

This note is intended to give a detailed proof of a theorem announced by the same author previously [Geometry of the Laplace operator, Honolulu/Hawaii 1979, Proc. Symp. Pure Math., Vol. 36, 91-146 (1980; Zbl 0461.58002), p. 140, Theorem 7.1]. The theorem says: Let \(X^ n\) be a closed normal pseudo-manifold with piecewise constant curvature metric. (i) If \(X^ n\) has nonnegative curvature, then \(X^ n\) is a real homology manifold. Moreover, the Betti number \(b^ i(X^ n)\leq \left( \begin{matrix} n\\ i\end{matrix} \right).\) (ii) If \(X^ n\) has positive curvature, then it is a real homology sphere.

This theorem as well as its proof are analogous to the vanishing theorem of S. Bochner [see Ann. Math., II. Ser. 49, 379-390 (1948; Zbl 0038.344); 50, 77-93 (1949; Zbl 0039.176)]; or S. I. Goldberg [Curvature and homology (1962; Zbl 0105.156), 89, Theorem 3.2.5 and a Corollary on the same page)]. Yet certain modifications are necessary due to appearance of singularities in the present case. We would like to mention that there are a few misprints in indices which may turn out to cause trouble for the careful readers.

This note is intended to give a detailed proof of a theorem announced by the same author previously [Geometry of the Laplace operator, Honolulu/Hawaii 1979, Proc. Symp. Pure Math., Vol. 36, 91-146 (1980; Zbl 0461.58002), p. 140, Theorem 7.1]. The theorem says: Let \(X^ n\) be a closed normal pseudo-manifold with piecewise constant curvature metric. (i) If \(X^ n\) has nonnegative curvature, then \(X^ n\) is a real homology manifold. Moreover, the Betti number \(b^ i(X^ n)\leq \left( \begin{matrix} n\\ i\end{matrix} \right).\) (ii) If \(X^ n\) has positive curvature, then it is a real homology sphere.

This theorem as well as its proof are analogous to the vanishing theorem of S. Bochner [see Ann. Math., II. Ser. 49, 379-390 (1948; Zbl 0038.344); 50, 77-93 (1949; Zbl 0039.176)]; or S. I. Goldberg [Curvature and homology (1962; Zbl 0105.156), 89, Theorem 3.2.5 and a Corollary on the same page)]. Yet certain modifications are necessary due to appearance of singularities in the present case. We would like to mention that there are a few misprints in indices which may turn out to cause trouble for the careful readers.

Reviewer: C.C.Hwang

### MSC:

53C20 | Global Riemannian geometry, including pinching |