. D G ] 8 J un 2 00 4 NON - REDUCTIVE HOMOGENEOUS PSEUDO - RIEMANNIAN MANIFOLDS OF DIMENSION FOUR
نویسندگان
چکیده
A method, due tó Elie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is simply connected, then it is shown to be diffeomorphic to IR 4. All metrics for the simply connected non-reductive Einstein spaces are given explicitly. There are no non-reductive pseudo-Riemannian homogeneous spaces of dimension two and none of dimension three with connected isotropy subgroup.
منابع مشابه
. D G ] 1 6 O ct 2 00 4 NON - REDUCTIVE HOMOGENEOUS PSEUDO - RIEMANNIAN MANIFOLDS OF DIMENSION FOUR
A method, due tó Elie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is s...
متن کاملACTION OF SEMISIMPLE ISOMERY GROUPS ON SOME RIEMANNIAN MANIFOLDS OF NONPOSITIVE CURVATURE
A manifold with a smooth action of a Lie group G is called G-manifold. In this paper we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of the isometries. The dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds...
متن کاملar X iv : m at h / 04 02 28 2 v 2 [ m at h . D G ] 5 A pr 2 00 4 COMPLETE CURVATURE HOMOGENEOUS PSEUDO - RIEMANNIAN MANIFOLDS
We exhibit 3 families of complete curvature homogeneous pseudo-Riemannian manifolds which are modeled on irreducible symmetric spaces and which are not locally homogeneous. All of the manifolds have nilpotent Jacobi operators; some of the manifolds are, in addition, Jordan Osserman and Jordan Ivanov-Petrova.
متن کاملar X iv : 0 80 6 . 16 32 v 2 [ m at h . D G ] 5 N ov 2 00 8 Geodesically complete Lorentzian metrics on some homogeneous 3 manifolds
In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non-unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentzian 3-manifolds with n...
متن کاملar X iv : m at h / 05 05 66 9 v 1 [ m at h . D G ] 3 1 M ay 2 00 5 INVARIANT RIEMANNIAN METRICS AND f - STRUCTURES ON FLAG MANIFOLDS
It turns out that if a reductive complement m of a homogeneous reductive space G/H possesses a number of properties (including its decomposability into an orthogonal sum of three Ad(H)-invariant irreducible subspaces) then there exists a simple way of determining whether an f -structure F on this space belongs to the classes G1f , NKf and Kill f of generalized Hermitian geometry with respect to...
متن کامل