On completeness of certain locally symmetric pseudo-Riemannian manifolds of signature $(2,2)$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912433471225856 |
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| author | Hanounah, Malek |
| author_facet | Hanounah, Malek |
| contents | 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 $Ω$ which is divisible by the action of a discrete group $Γ$ of $\operatorname{Isom}(\mathbf{X})$, i.e. $Γ$ acts properly and cocompactly on $Ω$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_13924 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On completeness of certain locally symmetric pseudo-Riemannian manifolds of signature $(2,2)$ Hanounah, Malek Differential Geometry 53C35, 53C50, 53C22, 57S30 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 $Ω$ which is divisible by the action of a discrete group $Γ$ of $\operatorname{Isom}(\mathbf{X})$, i.e. $Γ$ acts properly and cocompactly on $Ω$. 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. |
| title | On completeness of certain locally symmetric pseudo-Riemannian manifolds of signature $(2,2)$ |
| topic | Differential Geometry 53C35, 53C50, 53C22, 57S30 |
| url | https://arxiv.org/abs/2506.13924 |