Spectral geometry of eta-Einstein Sasakian manifolds

Park, JeongHyeong

doi:10.1016/j.geomphys.2012.06.007

Cited by 2 publications

(2 citation statements)

References 17 publications

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“…Regarding the spectral characterizations of Sasakian manifolds, in the recent paper [9] the author shows the following result. Let (M, g, η) and (M ′ , g ′ , η ′ ) be two compact Sasakian manifolds, dim M = m, dim M ′ = m ′ , m, m ′ ≥ 5.…”

Section: Introductionmentioning

confidence: 92%

A characterization of Sasakian space forms by the spectrum

Perrone

2015

Journal of Geometry and Physics

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Section: Introductionmentioning

confidence: 92%

A characterization of Sasakian space forms by the spectrum

Perrone

2015

Journal of Geometry and Physics

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“…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]. Curvature plays an important role in spectral geometry -see, for example, [2].…”

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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Spectral geometry of eta-Einstein Sasakian manifolds

Cited by 2 publications

References 17 publications

A characterization of Sasakian space forms by the spectrum

A characterization of Sasakian space forms by the spectrum

Universal Curvature Identities Iii

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