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MiS Preprint

A new proof of Cheeger-Gromoll soul conjecture and Takeuchi Theorem

Jianguo Cao and Mei-Chi Shaw


In this paper, we study the geometry for the evolution of (possibly non-smooth) equi-distant hypersurfaces in real and complex manifolds. First we use the matrix-valued Riccati equation to provide a new proof of the Takeuchi Theorem for peudo-convex Kähler domains with positive curvature. We derive a new monotone principle for both smooth and non-smooth portions of equi-distant hypersurfaces in manifolds with nonnegative curvature.

Such a new monotone principle leads to a new proof of the Cheeger-Gromoll soul conjecture without using Perelman's flat strip theorem.

In addition, we show that if $M^n$ is a complete, non-compact $C^\infty$-smooth Riemannian manifold with nonnegative sectional curvature, then any distance non-increasing retraction from $M^n$ to its soul $\mathcal{S}$ must be a $C^\infty$-smooth Riemannian submersion, a result obtained independently by B. Wilking.

MSC Codes:
non-negative curvature, cheeger-gromoll soul conjecture, oka lemma and takeuchi theorem, distance non-increasing retraction

Related publications

2003 Repository Open Access
Jianguo Cao and Mei-Chi Shaw

A new proof of Cheeger-Gromoll soul conjecture and Takeuchi Theorem