A warped product version of the cheeger-gromoll splitting theorem

Research output: Research - peer-reviewArticle

Abstract

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 non-negative 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).

LanguageEnglish (US)
Pages6661-6681
Number of pages21
JournalTransactions of the American Mathematical Society
Volume369
Issue number9
DOIs
StatePublished - 2017

Fingerprint

Warped Product
Theorem
Nonnegative Curvature
Sectional Curvature
Fundamental Group
Compact Manifold
Curvature
Warping
Riemannian Metric
Obstruction
Riemannian Manifold
Metric
Line
Form
Generalization

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

A warped product version of the cheeger-gromoll splitting theorem. / Wylie, William.

In: Transactions of the American Mathematical Society, Vol. 369, No. 9, 2017, p. 6661-6681.

Research output: Research - peer-reviewArticle

@article{1a4800463c144793801707a7b84bf1d1,
title = "A warped product version of the cheeger-gromoll splitting theorem",
abstract = "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 non-negative 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).",
author = "William Wylie",
year = "2017",
doi = "10.1090/tran/7003",
volume = "369",
pages = "6661--6681",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "9",

}

TY - JOUR

T1 - A warped product version of the cheeger-gromoll splitting theorem

AU - Wylie,William

PY - 2017

Y1 - 2017

N2 - 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 non-negative 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).

AB - 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 non-negative 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).

UR - http://www.scopus.com/inward/record.url?scp=85020471589&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85020471589&partnerID=8YFLogxK

U2 - 10.1090/tran/7003

DO - 10.1090/tran/7003

M3 - Article

VL - 369

SP - 6661

EP - 6681

JO - Transactions of the American Mathematical Society

T2 - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 9

ER -