Curvature Structure of Self-Dual 4-Manifolds

Blažić, Novica; Gilkey, Peter Β.; Nikčević, Stana; Stavrov, Iva

doi:10.1142/s0219887808003259

Cited by 8 publications

(3 citation statements)

References 23 publications

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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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Section: Introductionmentioning

confidence: 99%

Universal Curvature Identities Iii

Gilkey

Park

Sekigawa

2013

Int. J. Geom. Methods Mod. Phys.

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“…The study of conformally Osserman manifolds was started in [BG1] and then continued in [BG2,BGNSi,Gil,BGNSt]. Every Osserman manifold is conformally Osserman (which easily follows from the formula for the Weyl tensor and the fact that every Osserman manifold is Einstein), as also is every manifold locally conformally equivalent to an Osserman manifold.…”

Section: Introductionmentioning

confidence: 99%

Conformally Osserman manifolds of dimension 16 and a Weyl–Schouten theorem for rank-one symmetric spaces

Nikolayevsky

2011

Annali di Matematica

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A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unit tangent sphere at every point. Osserman Conjecture asserts that every Osserman manifold is either flat or rank-one symmetric. We prove that both the Osserman Conjecture and its conformal version, the Conformal Osserman Conjecture, are true, modulo a certain assumption on algebraic curvature tensors in R 16 . As a consequence, we show that a Riemannian manifold having the same Weyl tensor as a rank-one symmetric space, is conformally equivalent to it.

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“…The study of conformally Osserman manifolds was started in [BG1], and then continued in [BG2,BGNSi,G2,BGNSt]. Every Osserman manifold is conformally Osserman (which easily follows from the formula for the Weyl tensor and the fact that every Osserman manifold is Einstein), as also is every manifold locally conformally equivalent to an Osserman manifold.…”

Section: Introductionmentioning

confidence: 99%

Conformally Osserman manifolds

Nikolayevsky¹

2010

Pacific J. Math.

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Dedicated to the memory of Novica Blažić (1959 -2005), a remarkable mathematician and a wonderful person.Abstract. An algebraic curvature tensor is called Osserman if the eigenvalues of the associated Jacobi operator are constant on the unit sphere. A Riemannian manifold is called conformally Osserman if its Weyl conformal curvature tensor at every point is Osserman. We prove that a conformally Osserman manifold of dimension n = 3, 4, 16 is locally conformally equivalent either to a Euclidean space or to a rank-one symmetric space.

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Curvature Structure of Self-Dual 4-Manifolds

Cited by 8 publications

References 23 publications

Universal Curvature Identities Iii

Universal Curvature Identities Iii

Conformally Osserman manifolds of dimension 16 and a Weyl–Schouten theorem for rank-one symmetric spaces

Conformally Osserman manifolds

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