Lorentz manifolds modelled on a Lorentz symmetric space. (English) Zbl 0736.53056

Cahen, M.; Leroy, J.; Parker, M.; Tricerri, Franco; Vanhecke, Lieven

J. Geom. Phys. 7, No. 4, 571-581 (1990).

Let \((M,g)\) be a pseudo-Riemannian manifold and \((M_ 0,g_ 0)\) a pseudo-Riemannian symmetric space of the same dimension and signature. One says that \((M,g)\) is modelled on \((M_ 0,g_ 0)\) if, for any point \(\xi\in M\), there exists an isometry \(\phi_ \xi: T_ \xi M\rightarrow T_{u_ 0}M_ 0\) (\(u_ 0\) being a base point of \(M_ 0\)) such that \(\varphi^*_ \xi R_{0,u_ 0}=R_ \xi\). This means that \((M,g)\) has “the same curvature tensor” as \((M_ 0,g_ 0)\). The authors prove that if \((M_ 0,g_ 0)\) is Lorentzian and irreducible, then \((M_ 0,g_ 0)\) has constant sectional curvature and each Lorentzian \((M,g)\) modelled on \((M_ 0,g_ 0)\) hsa also constant sectional curvature. On the other hand, they give examples of 3-dimensional Lorentz manifolds \((M,g)\) modelled on an indecomposable Lorentz symmetric space \((M_ 0,g_ 0)\) which are geodesically complete and not locally homogeneous.

Reviewer: O.Kowalski (Praha)

MSC:

53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
53C30	Differential geometry of homogeneous manifolds
53C35	Differential geometry of symmetric spaces

Keywords:

curvature homogeneous space; pseudo-Riemannian symmetric space; constant sectional curvature; Lorentz symmetric space

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