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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
Published electronically: July 21, 2016
MathSciNet review: 3565389
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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.

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Amir Babak Aazami
Affiliation: Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

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

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