Conformal Ricci solitons and related integrability conditions

International Mathematics Research Notices

2018

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The very definition of an Einstein metric implies that all its geometry is encoded in the Weyl tensor. With this in mind, in this paper we derive higher-order Bochner type formulas for the Weyl tensor on a four dimensional Einstein manifold. In particular, we prove a second Bochner type formula which, formally, extends to the covariant derivative level the classical one for the Weyl tensor obtained by Derdzinski in 1983. As a consequence, we deduce some integral identities involving the Weyl tensor and its derivatives on a compact four dimensional Einstein manifold.

“…For the second and third derivatives of W , it is known that (see for instance [2]) Lemma 3.3. On every n-dimensional, n ≥ 4, Riemannian manifold one has…”

Section: Some Algebraic Formulas For the Weyl Tensormentioning

confidence: 99%

Bochner-type Formulas for the Weyl Tensor on Four-dimensional Einstein Manifolds

International Mathematics Research Notices

2018

Self Cite

“…A computation using the commutation rules for the second covariant derivative of the Weyl tensor or of the Schouten tensor (see [25]) shows that the Bach tensor is symmetric (i.e. B ij = B ji ); it is also evidently trace-free (i.e.…”

Section: Definitions and Some Useful Formulasmentioning

confidence: 99%

“…This definition follows from a previous work of the authors [25], where we derived the so called integrability conditions for nongradient Ricci solitons. Assume div [E X g] = 0.…”

Section: Nongradient Canonical Metricsmentioning

confidence: 99%

A potential generalization of some canonical Riemannian metrics

2019

Ann Glob Anal Geom

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The aim of this paper is to study new classes of Riemannian manifolds endowed with a smooth potential function, including in a general framework classical canonical structures such as Einstein, harmonic curvature and Yamabe metrics, and, above all, gradient Ricci solitons. For the most rigid cases we give a complete classification, while for the others we provide rigidity and obstruction results, characterizations and nontrivial examples. In the final part of the paper we also describe the "nongradient" version of this construction.2010 Mathematics Subject Classification. 53C20, 53C25. 1 2 GIOVANNI CATINO AND PAOLO MASTROLIAOn the other hand, from the analytic point of view, the aim is to simplify the curvature by imposing some differential condition. A quite natural and not too restrictive way to do this is to consider curvature tensors belonging to the kernel of a first order linear differential operator. Some well known conditions of this type can be given by saying that (M, g) belongs to• LS if ∇(Riem) = 0 (locally symmetric metrics); • PR if ∇(Ric) = 0 (metrics with parallel Ricci curvature); • HC if div(Riem) = 0 (harmonic curvature metric).Note that, by Bianchi identities, we can redefine the class Y of Yamabe metrics using the condition div(Ric) = 0 or, equivalently, ∇R = 0. Here and in the rest of the paper div denotes the divergence operator (see Section 3 for the definition). Obviously, SF ⊂ LS ⊂ PR and, by Bianchi identities, PR ⊂ HC, E ⊂ HC ⊂ Y. Thus, we have the inclusionswhere, by definition, LSE := LS ∩ E (locally symmetric Einstein metrics). Note that, in dimension n ≥ 4, all the inclusions are strict.This classes of metrics certainly do not exhaust all the possible canonical metrics on a Riemannian manifold: our choice is essentially made in such a way that Einstein metrics (and Ricci solitons, as we will see) are the "cornerstone" of our construction, and the conditions that we impose are consequently focused on the Ricci tensor. We note that, in principle, one could also consider "higher order" conditions, such as ∇ k Riem = 0 or ∇ k Ric = 0, k ≥ 2, but these relations give rise again to LS and PR, respectively, by the results in [46,50]. However, one can consider other higher order analytic curvature conditions in order to generalize locally symmetric metrics, such as, for instance, the class of semi-symmetric spaces introduced by Cartan in [19].The class SF is the most rigid, since, up to quotients, it contains only S n , R n and H n with their standard metrics. Locally symmetric spaces LS were classified by Cartan [18], while, from the de Rham decomposition theorem ([6]), PR metrics are locally Riemannian products of Einstein metrics. On the other hand, given any compact manifold M, there always exists a Riemannian metric g such that (M, g) ∈ Y (see e.g. [41]). The remaining classes are more flexible. In particular E and HC, in the last decades, have been studied by many researchers, also for their connections with Physics in General Relativity and Yang-Mills Theory. In fact, these metrics ...

“…by its skew-symmetry and Schur lemma. Furthermore, it satisfies C ijk,i = 0, see for instance [10,Equation 4.43]. We recall that, for n ≥ 4, the Cotton tensor can also be defined as one of the possible divergences of the Weyl tensor:…”

Section: Preliminariesmentioning

confidence: 99%

Weyl Scalars on Compact Ricci Solitons

2018

J Geom Anal

We investigate the triviality of compact Ricci solitons under general scalar conditions involving the Weyl tensor. More precisely, we show that a compact Ricci soliton is Einstein if a generic linear combination of divergences of the Weyl tensor contracted with suitable covariant derivatives of the potential function vanishes. In particular we recover and improve all known related results. This paper can be thought as a first, preliminary step in a general program which aims at showing that Ricci solitons can be classified finding a "generic" [k, s]-vanishing condition on the Weyl tensor, for every k, s ∈ N, where k is the order of the covariant derivatives of Weyl and s is the type of the (covariant) tensor involved.Ric +∇ 2 f = λg,