On the Sign of the Curvature of a Contact Metric Manifold
In this expository article, we discuss the author’s conjecture that an associated metric for a given contact form on a contact manifold of dimension ≥5 must have some positive curvature. In dimension 3, the standard contact structure on the 3-torus admits a flat associated metric...
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MDPI AG
2019-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/10/892 |
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doaj-5d4532eefb804e468cf8328f5c00e25b2020-11-25T02:45:11ZengMDPI AGMathematics2227-73902019-09-0171089210.3390/math7100892math7100892On the Sign of the Curvature of a Contact Metric ManifoldDavid E. Blair0Department of Mathematics, Michigan State University, East Lansing, MI 48824, USAIn this expository article, we discuss the author’s conjecture that an associated metric for a given contact form on a contact manifold of dimension ≥5 must have some positive curvature. In dimension 3, the standard contact structure on the 3-torus admits a flat associated metric; we also discuss a local example, due to Krouglov, where there exists a neighborhood of negative curvature on a particular 3-dimensional contact metric manifold. In the last section, we review some results on contact metric manifolds with negative sectional curvature for sections containing the Reeb vector field.https://www.mdpi.com/2227-7390/7/10/892contact manifoldsassociated metricscurvature |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
David E. Blair |
| spellingShingle |
David E. Blair On the Sign of the Curvature of a Contact Metric Manifold Mathematics contact manifolds associated metrics curvature |
| author_facet |
David E. Blair |
| author_sort |
David E. Blair |
| title |
On the Sign of the Curvature of a Contact Metric Manifold |
| title_short |
On the Sign of the Curvature of a Contact Metric Manifold |
| title_full |
On the Sign of the Curvature of a Contact Metric Manifold |
| title_fullStr |
On the Sign of the Curvature of a Contact Metric Manifold |
| title_full_unstemmed |
On the Sign of the Curvature of a Contact Metric Manifold |
| title_sort |
on the sign of the curvature of a contact metric manifold |
| publisher |
MDPI AG |
| series |
Mathematics |
| issn |
2227-7390 |
| publishDate |
2019-09-01 |
| description |
In this expository article, we discuss the author’s conjecture that an associated metric for a given contact form on a contact manifold of dimension ≥5 must have some positive curvature. In dimension 3, the standard contact structure on the 3-torus admits a flat associated metric; we also discuss a local example, due to Krouglov, where there exists a neighborhood of negative curvature on a particular 3-dimensional contact metric manifold. In the last section, we review some results on contact metric manifolds with negative sectional curvature for sections containing the Reeb vector field. |
| topic |
contact manifolds associated metrics curvature |
| url |
https://www.mdpi.com/2227-7390/7/10/892 |
| work_keys_str_mv |
AT davideblair onthesignofthecurvatureofacontactmetricmanifold |
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1724763669175205888 |