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...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2019-09-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/7/10/892 |
| _version_ | 1818048343941578752 |
|---|---|
| author | David E. Blair |
| author_facet | David E. Blair |
| author_sort | David E. Blair |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-12-10T10:20:11Z |
| format | Article |
| id | doaj.art-5d4532eefb804e468cf8328f5c00e25b |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-10T10:20:11Z |
| publishDate | 2019-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-5d4532eefb804e468cf8328f5c00e25b2022-12-22T01:52:53ZengMDPI 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 |
| spellingShingle | David E. Blair On the Sign of the Curvature of a Contact Metric Manifold Mathematics contact manifolds associated metrics curvature |
| title | 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_short | On the Sign of the Curvature of a Contact Metric Manifold |
| title_sort | on the sign of the curvature of a contact metric manifold |
| topic | contact manifolds associated metrics curvature |
| url | https://www.mdpi.com/2227-7390/7/10/892 |
| work_keys_str_mv | AT davideblair onthesignofthecurvatureofacontactmetricmanifold |