On completeness of certain locally symmetric pseudo-Riemannian manifolds of signature $(2,2)$

TL;DR

This paper proves geodesic completeness for certain compact, locally symmetric pseudo-Riemannian manifolds of signature (2,n), extending known results and classifying specific geometric structures.

Contribution

It establishes geodesic completeness for a new class of signature (2,n) manifolds, generalizing Cahen–Wallach spaces, and classifies related Kleinian manifolds.

Findings

Geodesic completeness for compact locally symmetric spaces modeled on these manifolds.

No proper divisible domains exist in the 4D case, implying completeness.

Classification of Kleinian compact manifolds on hyperbolic oscillator groups.

Abstract

We show geodesic completeness of certain compact locally symmetric pseudo-Riemannian manifolds of signature $(2,n)$. Our model space $\mathbf{X}$ is a $1$-connected, indecomposable symmetric space of signature $(2,n)$, that admits a unique (up to scale) parallel lightlike vector field. This class of spaces is the natural generalization of the class of Cahen--Wallach spaces to signature $(2,n)$. In dimension $4$ we show that $\mathbf{X}$ has no proper domain $\Omega$ which is divisible by the action of a discrete group $\Gamma$ of $\operatorname{Isom}(\mathbf{X})$, i.e. $\Gamma$ acts properly and cocompactly on $\Omega$. Therefore, we deduce geodesic completeness in the aforementioned situation. In arbitrary dimension we show geodesic completeness of compact locally symmetric space modeled on $\mathbf{X}$ under the assumption that the transition maps of $M$ are restrictions of transvections of $\mathbf{X}$. Along the way, we establish a new case in the Kleinian $3$-dimensional Markus's conjecture for flat affine manifolds. Moreover, we classify geometrically Kleinian compact manifolds that are modeled on the hyperbolic oscillator group endowed with its bi-invariant metric. Finally we discuss a natural dynamical problem motivated by the Lorentz setting (Brinkmann spacetimes). Specifically, we show that the parallel flow on $M$ is equicontinuous in dimension $4$, even in our non-Lorentz setting.