On complete manifolds of nonnegative \(k\)th-Ricci curvature. (English) Zbl 0783.53026

Shen, Zhongmin

Trans. Am. Math. Soc. 338, No. 1, 289-310 (1993).

The author studies the \(k\)-Ricci curvature, which is a kind of an interpolation between sectional and Ricci curvature. He obtains results in the spirit of theorems of J. Cheeger, D. Gromoll, W. Meyer, J. Sha and H. Wu. In particular, a complete open manifold of positive \(k\)-Ricci curvature and proper Busemann function has the homotopy type of a \((k- 1)\)-dimensional CW complex. The author verifies the condition on the Busemann function in the case of small diameter growth at infinity. The author also proves bounds on the Betti numbers in the case of nonnegative Ricci curvature in the spirit of Gromov and Abresch-Gromoll.

Reviewer: M.Katz (Vandoeuvre)

MSC:

53C20	Global Riemannian geometry, including pinching
31C12	Potential theory on Riemannian manifolds and other spaces

Keywords:

Busemann function; CW complex; Betti numbers