Lorentzian 3-Manifolds With Commuting Curvature Operators

Garcı́a-Rı́o, Eduardo; Haji‐Badali, Ali; Abal, María Elena Vázquez; Vázquez-Lorenzo, Ramón

doi:10.1142/s0219887808002941

Cited by 11 publications

(9 citation statements)

References 17 publications

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“…When (i, j, k, h) = (1, 2, 1, 2 16) and (2.20) it follows at once that the Ricci operator of any semi-symmetric Lorentzian manifold, curvature homogeneous up to order one, is either two-step nilpotent, or diagonalizable with eigenvalues 0, b, b. This algebraic condition characterizes the class of Lorentzian three-manifolds with commuting curvature and Ricci operator, studied in [17].…”

Section: Semi-symmetric Lorentzian Metrics In the Classes M 1 And Mmentioning

confidence: 96%

Semi-symmetric Lorentzian metrics and three-dimensional curvature homogeneity of order one

Calvaruso

2009

Abh. Math. Semin. Univ. Hambg.

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Section: Semi-symmetric Lorentzian Metrics In the Classes M 1 And Mmentioning

confidence: 96%

Semi-symmetric Lorentzian metrics and three-dimensional curvature homogeneity of order one

Calvaruso

2009

Abh. Math. Semin. Univ. Hambg.

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“…where {u, v}, u = u i ∂ i , v = v i ∂ i , is a positively oriented orthonormal basis of π and Ξ = |g(u, u)g(v, v) − g(u, v) 2 | 1/2 . Now, (10) implies that the eigenvalues of the skew-symmetric curvature operator vanish identically (in particular, R(π) is again three-step nilpotent). Hence, we have the following…”

Section: On the Curvature Of Egorov Spaces And ε-Spacesmentioning

confidence: 99%

“…For several of these properties, a complete description of the corresponding manifolds has not been obtained yet. A recent survey can be found in [3], and we can refer to [10] for the sistematic study of the three-dimensional Lorentzian case.…”

Section: Commuting Curvature Operators and Semi-symmetrymentioning

confidence: 99%

Algebraic Properties of Curvature Operators in Lorentzian Manifolds with Large Isometry Groups

Calvaruso

Garcı́a-Rı́o

2010

SIGMA

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Abstract. Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and ε-spaces exhaust the class of n-dimensional Lorentzian manifolds admitting a group of isometries of dimension at least 1 2 n(n − 1) + 1, for almost all values of n [Patrangenaru V., Geom. Dedicata 102 (2003), 25-33]. We shall prove that the curvature tensor of these spaces satisfy several interesting algebraic properties. In particular, we will show that Egorov spaces are IvanovPetrova manifolds, curvature-Ricci commuting (indeed, semi-symmetric) and P-spaces, and that ε-spaces are Ivanov-Petrova and curvature-curvature commuting manifolds.

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“…The study of curvature is central to modern differential geometry and mathematical physics. Properties of the curvature operator have been examined by many authors -see, for example, the discussion in [4,12]. Eta Einstein geometry has been investigated [10,24].…”

Section: Introductionmentioning

confidence: 99%

Universal Curvature Identities Iii

Gilkey

Park

Sekigawa

2013

Int. J. Geom. Methods Mod. Phys.

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We examine universal curvature identities for pseudo-Riemannian manifolds with boundary. We determine the Euler-Lagrange equations associated to the Chern-Gauss-Bonnet formula and show that they are given solely in terms of curvature and the second fundamental form and do not involve covariant derivatives, thus generalizing a conjecture of Berger to this context. The Gauss-Bonnet theorem has many physical applications as are the associated Euler-Lagrange equations (see, for example, [5,17,24]). This paper deals with universal curvature identities arising from the Euler-Lagrange equations for the Chern-Gauss-Bonnet theorem for manifolds with boundary. In Secs. 1.1 and 1.2, we discuss the Gauss-Bonnet theorem for closed pseudo-Riemannian manifolds, present the associated Euler-Lagrange equations, and discuss some of the historical context of the problem that we shall be considering. In Sec. 1.3, we recall the Gauss-Bonnet theorem for manifolds with boundary; we shall always assume the restriction of the pseudo-Riemannian metric to the boundary to be non-degenerate. In Sec. 1.4, we state Theorem 1.4 -this is the first result of the paper. It gives the associated Euler-Lagrange equations for the Gauss-Bonnet integrand for manifolds with boundary.The remainder of the paper is devoted to the proof of Theorem 1.4. Section 2 treats basic invariance theory; the question of universal curvature identities is central. In Theorem 2.2, we shall summarize previous results concerning universal curvature identities in the scalar case (both in the interior and on the boundary) and in the symmetric 2-tensor case in the interior. Theorem 2.3 is the second main result of this paper. In it, we extend the results of Theorem 2.2 to discuss universal curvature identities for symmetric 2-tensors defined by the geometry of the embedding ∂M ⊂ M . There is a technical fact we shall need in the proof of Theorem 2.3 that we postpone until Sec. 4 to avoid breaking the flow of the discussion. In Sec. 3, we use Theorem 2.3 to complete the proof of Theorem 1.4.Section 4 provides the technical results which are central to the discussion. Rather than using Weyl's theory of invariants [25] to treat the universal curvature identities which arise in Theorem 1.4, we have chosen to adopt the approach of [12] which was originally used to give a heat equation proof of the Gauss-Bonnet theorem. This seemed easier rather than having to introduce an additional complicated formalism to use results of [25]. The Gauss-Bonnet theorem for closed manifoldsLet (M, g) be a compact pseudo-Riemannian manifold of signature (p, q) and dimension m = p + q with smooth boundary ∂M ; if the signature is indefinite, we assume the restriction of the metric to the boundary to be non-degenerate. Let dx g be the Riemannian element of volume. Let e := {e 1 , . . . , e m } be a local orthonormal frame for the tangent bundle of M . Set ε j1...jn i1...in = ε(g, e ) j1...jn i1...in := g(e i1 ∧ · · · ∧ e in , e j1 ∧ · · · ∧ e jn ) = det(g(e iµ , e jν )) = ±1.(1.1)Clearly this vanishes ...

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Lorentzian 3-Manifolds With Commuting Curvature Operators

Abstract: Three-dimensional Lorentzian manifolds with commuting curvature operators are studied. A complete description is given at the algebraic level. Consequences are obtained at the differentiable setting for manifolds which additionally are assumed to be locally symmetric or homogeneous.

Cited by 11 publications

References 17 publications

Semi-symmetric Lorentzian metrics and three-dimensional curvature homogeneity of order one

Semi-symmetric Lorentzian metrics and three-dimensional curvature homogeneity of order one

Algebraic Properties of Curvature Operators in Lorentzian Manifolds with Large Isometry Groups

Universal Curvature Identities Iii

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