##### Department of Mathematics,

University of California San Diego

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### Math 258 - Differential Geometry

## Amir Babak Aazami

## 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 $g(X,â€¢)$ 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 $dg(X,â€¢)$ is a symplectic form and X is a Liouville vector field.

Host: Lei Ni

### May 14, 2015

### 1:00 PM

### AP&M 5402

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