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Kleinian groups and conformally flat metrics. (English) Zbl 1040.53502
Kotake, Takeshi (ed.) et al., Geometry and global analysis. Report of the 1st MSJ (Mathematical Society of Japan) International Research Institute, Sendai, Japan, July 12–23, 1993. Sendai: Tôhoku Univ., Mathematical Institute. 341-349 (1993).
From the text: This paper is a supplement to another paper by the author [Math. Z. 225, 115–131 (1997; Zbl 0868.53024)]. After restating the main results from that paper, the author gives a new cohomology vanishing theorem stating that on a compact conformally flat Riemannian manifold, if the sum of the smallest \(k\) eigenvalues of the Schouten tensor \((n-2)^{-1} \{\text{Ric}_g - (2n-2)^{-1} S_g g \}\) is everywhere positive, then the Betti numbers \(b_p\) of the manifold vanish for \( k/2 \leq p \leq n-k/2. \) As a corollary, a conformally flat closed Riemannian manifold with positive isotropic curvature has vanishing \(b_p, 2\leq p \leq n-2.\)
For the entire collection see [Zbl 0895.00042].

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
53C20 Global Riemannian geometry, including pinching