Maximal simply connected Lorentzian surfaces with a Killing field and their completeness

Completeness of certain compact Lorentzian locally symmetric spaces

“…if (M , g ) admits a parallel null vector field and satisfies a certain curvature condition. Very recently this result was generalised by Mehidi and Zeghib [10], who showed that the curvature condition can be dropped, i.e. that the existence of a parallel null vector field on a compact Lorentzian manifold implies that the manifold is complete.…”

“…In the second author's thesis [11] this was generalised to locally symmetric spaces which are local products of Euclidean space and a Cahen-Wallach space. Using the same method as in [11], the new result in [10], as well as the result in [15], here we generalise this further.…”

“…If we now assume that M is compact, the time-orientable cover is compact and admits a global parallel null vector field, and hence, by the result in [10] is geodesically complete. This implies that (M , g ) is geodesically complete.…”

On completeness and dynamics of compact Brinkmann spacetimes

“…Generic Lorentz metrics on compact manifolds are thought of to be incomplete! The historical example is the Clifton-Pohl torus R 2 − {(0, 0)}/ (x,y)∼2(x,y) endowed with the metric dxdy x 2 +y 2 (see [13,34,35,38,28] for various results on completeness of Lorentz surfaces).…”