Skip to main content Accessibility help
×
Home

Second Variation of the "Total Scalar Curvature" on Contact Manifolds

  • D. E. Blair (a1) and D. Perrone (a2)

Abstract

Let M 2n+1 be a compact contact manifold and 𝓐 the set of associated metrics. Using the scalar curvature R and the *-scalar curvature R*, in [5] we defined the "total scalar curvature", by and showed that the critical points of I(g) on 𝓐 are the K-contact metrics, i.e. metrics for which the characteristic vector field is Killing. In this paper we compute the second variation of I(g) and prove that the index of I(g) and of —I(g) are both positive at each critical point. As an application we show that the classical total scalar curvature A(g) = ∫M R dVg restricted to 𝓐 cannot have a local minimum at any Sasakian metric.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Second Variation of the "Total Scalar Curvature" on Contact Manifolds
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Second Variation of the "Total Scalar Curvature" on Contact Manifolds
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Second Variation of the "Total Scalar Curvature" on Contact Manifolds
      Available formats
      ×

Copyright

References

Hide All
1. Blair, D. E., Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. 509, Springer, Berlin, 1976.
2. Blair, D. E., On the set of metrics associated to a symplectic or contact form, Bull. Inst. Math. Acad. Sinica 11(1983), 297308.
3. Blair, D. E., The “total scalar curvature“as a symplectic invariant, Proc. 3rd Congress of Geometry, Thessaloniki, 1991,79-83.
4. Blair, D. E. and Ledger, A. J., Critical associated metrics on contact manifolds II, J. Austral. Math. Soc. Ser. A 41(1986), 404410.
5. Blair, D. E. and D. Perrone, A variational characterization of contact metric manifolds with vanishing torsion, Canad. Math. Bull., 35(1992), 455462.
6. Chern, S. S. and Hamilton, R. S., On Riemannian metrics adapted to three-dimensional contact manifolds, Lecture Notes in Math. 1111, Springer, Berlin, 1985, 279308.
7. Muto, Y., On Einstein metrics, J. Differential Geom. 9(1974), 521530.
8. Olszak, Z, On contact metric manifolds, Tôhoku Math. J. 31(1979), 247253.
9. Perrone, D., Torsion and critical metrics on contact three-manifolds, Kodai Math. J. 13(1990), 88100.
10. Perrone, D., Torsion tensor and critical metrics on contact (2n+ \)-manifolds, Mh. Math. 114(1992), 245259.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Second Variation of the "Total Scalar Curvature" on Contact Manifolds

  • D. E. Blair (a1) and D. Perrone (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed