关键词: 球定理, 广义Poincar\''e猜想, 第$k$个Ricci曲率, 同胚

Abstract:

This paper discusses the topology of manifolds with a small well of positive $k$-th Ricci curvature and generalizes the classical sphere theorem. Also, the authors will describe an interesting compensatory phenomenon of the manifold with respect to the classical sphere theorem.

Key words: Sphere theorem, Generalized Poincar\''{e} conjecture, $k$-th Ricci curvature, Homeormorphism