Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 

 

Symplectic 4-manifolds via Lorentzian geometry


Author: Amir Babak Aazami
Journal: Proc. Amer. Math. Soc. 145 (2017), 387-394
MSC (2010): Primary 53C50; Secondary 53D05
DOI: https://doi.org/10.1090/proc/13226
Published electronically: July 21, 2016
MathSciNet review: 3565389
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We observe that, in dimension four, symplectic forms may be obtained via Lorentzian geometry; in particular, null vector fields can give rise to exact symplectic forms. That a null vector field is nowhere vanishing yet orthogonal to itself is essential to this construction. Specifically, we show that on a Lorentzian 4-manifold $ (M,g)$, if $ \boldsymbol {k}$ is a complete null vector field with geodesic flow along which $ \text {Ric}(\boldsymbol {k},\boldsymbol {k})>0$, and if $ f$ is any smooth function on $ M$ with $ \boldsymbol {k}(f)$ nowhere vanishing, then $ dg(e^f\boldsymbol {k},\cdot )$ is a symplectic form and $ \boldsymbol {k}/\boldsymbol {k}(f)$ is a Liouville vector field; any null surface to which $ \boldsymbol {k}$ is tangent is then a Lagrangian submanifold. Even if the Ricci curvature condition is not satisfied, one can still construct such symplectic forms with additional information from $ \boldsymbol {k}$. We give an example of this, with $ \boldsymbol {k}$ a complete Liouville vector field, on the maximally extended ``rapidly rotating'' Kerr spacetime.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C50, 53D05

Retrieve articles in all journals with MSC (2010): 53C50, 53D05


Additional Information

Amir Babak Aazami
Affiliation: Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
Email: amir.aazami@ipmu.jp

DOI: https://doi.org/10.1090/proc/13226
Received by editor(s): October 19, 2015
Received by editor(s) in revised form: February 26, 2016, and March 28, 2016
Published electronically: July 21, 2016
Communicated by: Guofang Wei
Article copyright: © Copyright 2016 American Mathematical Society