A warped product version of the Cheeger-Gromoll splitting theorem
By W. Wylie. 2015
57 citations recorded by the source corpus.
We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form $CD(0,1)$. Even though we have to allow warping in our splitting, we are able to recover topological applications. In particular, for a smooth compact Riemannian manifold admitting a density which is $CD(0,1)$, we show that the fundamental group of $M$ is the fundamental group of a compact manifold with nonnegative sectional curvature. If the space is also locally homogeneous, we obtain that the space also admits a metric of non-negative sectional curvature. Both of these obstructions give many examples of Riemannian metrics which do not admit any smooth density which is $CD(0,1)$.
Subfields: Applied Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/9d0df5800f6e8b83c397198fdd7e5543c28adea1
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In Applied Mathematics
- Emanuel Milman
- Shin-ichi Ohta
- A. V. Kolesnikov
- Fabio Cavalletti
- Andrea Mondino
- Karl‐Theodor Sturm
- Nicola Gigli
- T. Rajala
In Geometry and Topology
- Emanuel Milman
- Shin-ichi Ohta
- A. V. Kolesnikov
- Nicola Gigli
- Giuseppe Savar'e
- Andrea Mondino
- Fabio Cavalletti
- Luigi Ambrosio
Citations in the corpus
- Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension (2014, methodology): “Finally, we mention the recent works by Wylie [63,64] and Kennard–Wylie [29] on generalizing the notion of sectional curvature to the weighted manifold setting and its applications.”
- Needle decompositions and isoperimetric inequalities in Finsler geometry (2015, background): “ the range of N was restricted in [n,∞] (see [Bak, Qi, Lo]). The case of N∈(−∞,0) was investigated rather recently in [OT1, OT2, MR, KM, Oh8, Mi2], and some results admit N ∈[0,1) and even N = 1 (see [Wy3]). Klartag’s work [Kl] also covers N∈(−∞,1) ∪[n,∞]. See [Mi3] for a recent interesting example ”