Department of Mathematics - UC Santa Barbara

Geometry, Topology, and Physics Seminar, Spring 2015

Part of the NSF/UCSB ‘Research Training Group’ in Topology and Geometry

Organizers: Dave Morrison and Zhenghan Wang.
Meets 4:00 - 5:30 p.m. on selected Fridays in South Hall 6635.

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May. 15

Amir Aazami (Kavli IPMU, Univ. of Tokyo)

Contact and symplectic structures on closed Lorentzian manifolds

Abstract: We investigate timelike and null vector flows on closed Lorentzian manifolds and their relationship to Ricci curvature. The guiding observation, first observed for closed Riemannian 3-manifolds by Harris & Paternain '13, is that positive Ricci curvature tends to yield contact forms, namely, 1-forms metrically equivalent to unit vector fields with geodesic flow. We carry this line of thought over to the Lorentzian setting. First, we observe that the same is true on a closed Lorentzian 3-manifold: if X is a global timelike unit vector field with geodesic flow satisfying Ric(X,X) > 0, then g(X,•) is a contact form with Reeb vector field X, at least one of whose integral curves is closed. Second, we show that on a closed Lorentzian 4-manifold, if X is a global null vector field satisfying \nabla_XX = X and Ric(X) > divX - 1, then dg(X,•) is a symplectic form and X is a Liouville vector field.