Ricci Curvature on Alexandrov spaces and Rigidity Theorems

@article{Zhang2009RicciCO,
  title={Ricci Curvature on Alexandrov spaces and Rigidity Theorems},
  author={Huichun Zhang and Xiping Zhu},
  journal={arXiv: Differential Geometry},
  year={2009}
}
In this paper, we introduce a new notion for lower bounds of Ricci curvature on Alexandrov spaces, and extend Cheeger-Gromoll splitting theorem and Cheng's maximal diameter theorem to Alexandrov spaces under this Ricci curvature condition. 

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References

SHOWING 1-10 OF 44 REFERENCES
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
Differential Geometric Aspects of Alexandrov Spaces
We summarize the results on the differential geometric structure of Alexandrov spaces developed in [Otsu and Shioya 1994; Otsu 1995; Otsu and Tanoue a]. We discuss Riemannian and second
Alexandrov meets Lott-Villani-Sturm
Here I show the compatibility of two definitions of generalized curvature bounds: the lower bound for sectional curvature in the sense of Alexandrov and the lower bound for Ricci curvature in the
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
Optimal transport and Ricci curvature in Finsler geometry
This is a survey article on recent progress (in [Oh3], [OS]) of the theory of weighted Ricci curvature in Finsler geometry. Optimal transport theory plays an impressive role as is developed in the
Harmonic functions on Alexandrov spaces and their applications
The main result can be stated roughly as follows: Let M be an Alexandrov space, Ω ⊂M an open domain and f : Ω→ R a harmonic function. Then f is Lipschitz on any compact subset of Ω. Using this result
On local Poincaré via transportation
It is shown that curvature-dimension bounds CD(N,K) for a metric measure space (X,d,m) in the sense of Sturm imply a weak L1-Poincaré-inequality provided (X,d) has m-almost surely no branching points.
A splitting theorem for Alexandrov spaces
We use a notion of differentiability for functions on Alexandrov spaces and prove a splitting theorem for Alexandrov spaces admitting affine functions with such differentiability.
Finsler interpolation inequalities
We extend Cordero-Erausquin et al.’s Riemannian Borell–Brascamp–Lieb inequality to Finsler manifolds. Among applications, we establish the equivalence between Sturm, Lott and Villani’s
Semiconcave Functions in Alexandrov???s Geometry
The following is a compilation of some techniques in Alexandrov's geometry which are directly connected to convexity.
...
...