Universal curvature identities II

Gilkey, Peter Β.; Park, J.H.; Sekigawa, Kouei

doi:10.1016/j.geomphys.2012.01.002

Cited by 17 publications

(16 citation statements)

References 27 publications

(54 reference statements)

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“…We demonstrated that the obtained curvature identity holds on any 4-dimensional Riemannian manifold which is not necessarily compact [11]. Further, Gilkey, Park and Sekigawa extended the result to the higher dimensional setting, the pseudo-Riemannian setting, manifolds with boundary setting and the Kähler setting [13,14,15,16]. In this paper, we shall give a curvature identity explicitly which holds on any 6-dimensional Riemannian manifold using methods similar to those used in the 4-dimensional Chern-Gauss-Bonnet theorem and also provide some applications of the obtained curvature identity.…”

Section: Introductionmentioning

confidence: 73%

A curvature identity on a~6-dimensional Riemannian manifold and its applications

2017

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Section: Introductionmentioning

confidence: 73%

A curvature identity on a~6-dimensional Riemannian manifold and its applications

2017

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“…If P n,C is a universal scalar curvature identity in the curvature tensor in signature (p, q), then P n,C is a universal scalar curvature in any other signature (p,q) if p + q =p +q. In the discussion of [17], we first passed to the algebraic setting and then used analytic continuation. The reason to avoid dealing with the covariant derivatives of the curvature tensor was the relation…”

Section: Spaces Of Invariantsmentioning

confidence: 99%

“…Furthermore, it is now clear that the restriction of a universal curvature identity in dimension m generates a corresponding universal curvature identity in dimension m − 1. One has [13,15,17]: Theorem 2.3. Adopt the notation established above.…”

Section: Spaces Of Invariantsmentioning

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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“…This involves, of course, examining the associated Euler-Lagrange equations for this functional. Indefinite signatures are also important; Gilkey, Park and Sekigawa examined the Euler-Lagrange equations in the higher signature setting [10].…”

Section: Introductionmentioning

confidence: 99%

Euler–Lagrange formulas for pseudo-Kähler manifolds

Park

2016

Journal of Geometry and Physics

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Let c be a characteristic form of degree k which is defined on a Kähler manifold of real dimension m > 2k. Taking the inner product with the Kähler form Ω k gives a scalar invariant which can be considered as a generalized Lovelock functional. The associated Euler-Lagrange equations are a generalized Einstein-Gauss-Bonnet gravity theory; this theory restricts to the canonical formalism if c = c 2 is the second Chern form. We extend previous work studying these equations from the Kähler to the pseudo-Kähler setting.Date: 26Apr15. MSC 2010: 53B35, 57R20

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Universal curvature identities II

Cited by 17 publications

References 27 publications

A curvature identity on a~6-dimensional Riemannian manifold and its applications

A curvature identity on a~6-dimensional Riemannian manifold and its applications

Universal Curvature Identities Iii

Euler–Lagrange formulas for pseudo-Kähler manifolds

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