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

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.

### MSC:

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

### Keywords:

Busemann function; CW complex; Betti numbers
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