Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-Emery Ricci curvature

@article{Fang2007TwoGO,
  title={Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-Emery Ricci curvature},
  author={Fuquan Fang and Xiangdong Li and Zhenlei Zhang},
  journal={Annales de l'Institut Fourier},
  year={2007},
  volume={59},
  pages={563-573}
}
Dans cet article, nous obtenons deux generalisations du theoreme de scindage de Cheeger-Gromoll sur les varietes riemanniennes completes a courbure de Ricci non-negative au sens de Bakry-Emery. 
A topological splitting theorem for weighted Alexandrov spaces
Under an infinitesimal version of the Bishop-Gromov relative volume comparison condition for a measure on an Alexandrov space, we prove a topological splitting theorem of Cheeger-Gromoll type. As a
Splitting theorems on complete manifolds with Bakry-Émery curvature
In this paper we study some splitting properties on complete noncompact manifolds with smooth measures when $\infty$-dimensional Bakry-\'Emery Ricci curvature is bounded from below by some negative
Splitting theorems for Finsler manifolds of nonnegative Ricci curvature
We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-Gromoll-Lichnerowicz splitting theorem. Such a
Degeneration of Shrinking Ricci Solitons
Let (Y, d) be a Gromov-Hausdorff limit of closed shrinking Ricci solitons with uniformly upper bounded diameter and lower bounded volume. We prove that off a closed subset of codimension at least 2,
Rigidity of manifolds with boundary under a lower Bakry-Émery Ricci curvature bound
We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude
Height estimates and half-space type theorems in weighted product spaces with nonnegative Bakry–Émery–Ricci curvature
We prove height estimates concerning compact hypersurfaces with nonzero constant weighted mean curvature and whose boundary is contained into a slice of a weighted product space of nonnegative
On $$\kappa $$κ-noncollapsed complete noncompact shrinking gradient Ricci solitons which split at infinity
We discuss some geometric conditions under which a complete noncompact shrinking gradient Ricci soliton will split at infinity.
A semigroup approach to Finsler geometry: Bakry--Ledoux's isoperimetric inequality
We develop the celebrated semigroup approach a la Bakry et al on Finsler manifolds, where natural Laplacian and heat semigroup are nonlinear, based on the Bochner-Weitzenbock formula established by
The Splitting Theorem and topology of noncompact spaces with nonnegative N-Bakry Émery Ricci curvature
In this paper, we generalize topological results known for noncompact manifolds with nonnegative Ricci curvature to spaces with nonnegative $N$-Bakry Emery Ricci curvature. We study the Splitting
Fundamental Groups of Spaces with Bakry–Emery Ricci Tensor Bounded Below
We first extend Cheeger–Colding’s Almost Splitting Theorem (Ann Math 144:189–237, 1996) to smooth metric measure spaces. Arguments utilizing this extension show that if a smooth metric measure space
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 50 REFERENCES
An elementary proof of the Cheeger-Gromoll splitting theorem
We give a short proof of the Cheeger-Gromoll Splitting Theorem which says that a line in a complete manifold of nonnegative Ricci curvature splits off isometrically. Our proof avoids the existence
Volume comparison theorems without Jacobi fields
Using a generalized curvature-dimension inequality and a new approach, we preset a differential inequality for an elliptic second order differential operator acting on distance functions, from which
Complete Shrinking Ricci Solitons have Finite Fundamental Group
We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound,
Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator
Abstract. We establish, by simple semigroup arguments, a Lévy–Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with
Etude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée
© Springer-Verlag, Berlin Heidelberg New York, 1987, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l’accord avec
Some geometric properties of the Bakry-Émery-Ricci tensor
Abstract The Bakry-Émery tensor gives an analog of the Ricci tensor for a Riemannian manifold with a smooth measure. We show that some of the topological consequences of having a positive or
Comparison geometry for the Bakry-Emery Ricci tensor
For Riemannian manifolds with a measure (M, g, edvolg) we prove mean curvature and volume comparison results when the ∞-Bakry-Emery Ricci tensor is bounded from below and f is bounded or ∂rf is
The Comparison Geometry of Ricci Curvature
We survey comparison results that assume a bound on the manifold’s Ricci curvature.
Harnack inequalities on a manifold with positive or negative Ricci curvature
Several new Harnack estimates for positive solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded below by a positive (or a negative) constant are established.
A remark on compact Ricci solitons
It is shown that a complete shrinking soliton is compact if and only if it is bounded and moreover, in such a case, it has finite fundamental group.
...
1
2
3
4
5
...