Communications in Analysis and Geometry

Volume 13 (2005)

Number 4

A connectedness principle in the geometry of positive curvature

Pages: 671 – 695

DOI: http://dx.doi.org/10.4310/CAG.2005.v13.n4.a2

Authors

Fuquan Fang

Sérgio Mendonça

Xiaochun Rong

Abstract

The main purpose of this paper is to develop a connectedness principle in the geometry of positive curvature. In the form, this is a surprising analog of the classical connectedness principle in complex algebraic geometry. The connectedness principle, when applied to totally geodesic immersions, provides not only a uniform formulation for the classical Synge theorem, the Frankel theorem and a recent theorem of Wilking for totally geodesic submanifolds, but also new connectedness theorems for totally geodesic immersions in the geometry of positive curvature. However, the connectedness principle may apply in certain cases which do not require the existence of totally geodesic immersions.

2010 Mathematics Subject Classification

53C21

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