On conformally flat manifolds with constant positive scalar curvature

Catino, Giovanni

doi:10.1090/proc/12925

Cited by 26 publications

(34 citation statements)

References 27 publications

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“…then 1)g is a Yamabe minimizer and (M 4 ,g) is the manifold which is isometrically covered by S 1 ×S 3 with the product metric or the manifold which is isometrically covered by S 1 ×S 3 with a rotationally symmetric Derdziński metric (see [9,14]); 2) (M 4 , g) is isometric to a quotient of S 2 × S 2 with the product metric.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Four-manifolds with positive Yamabe constant

2018

Pacific J. Math.

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We refine Theorem B due to Gursky [26]. As applications, we give some rigidity theorems on four-manifolds with postive Yamabe constant. In particular, these rigidity theorems are sharp for our conditions have the additional properties of being sharp. By this we mean that we can precisely characterize the case of equality. We prove some classification theorems of four manifolds according to some conformal invariants (see Theorems 5.1, 1.3 and 1.6), which reprove and generalize the conformally invariant sphere theorem of Chang-Gursky-Yang [11], i.e., Theorem D.2000 Mathematics Subject Classification. Primary 53C21; Secondary 53C20.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Four-manifolds with positive Yamabe constant

2018

Pacific J. Math.

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“…is a trace-free Codazzi tensor. In particular, we have the following regularity lemma which follows from a general results of Kazdan [37] (see also [35,20] for some applications). Using this, together with Lemma 6.1, one has that either…”

Section: The Class Hc F : Rigidity Results Characterizations and Examentioning

confidence: 96%

“…Then • Ric f is a trace-free Codazzi tensor. In particular (see [6] or [20]), the following Weitzenböck formula holds Two examples. We construct two examples of Riemannian manifolds in HC f , following the construction for the harmonic curvature case given by Derdzinski in [27], following the same notation to highlight the similarities.…”

Section: The Class Hc F : Rigidity Results Characterizations and Examentioning

confidence: 99%

A potential generalization of some canonical Riemannian metrics

Catino

Mastrolia

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 ...

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“…Some of these results can be found in the monograph [2] which was published in 1987. On the other hand, there are many papers on the geometry of Codazzi tensors [5] - [10] which were published in subsequent years.…”

Section: Introductionmentioning

confidence: 99%

On higher-order Codazzi tensors on complete Riemannian manifolds

2019

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We prove several Liouville-type non-existence theorems for higher order Codazzi tensors and classical Codazzi tensors on complete and compact Riemannian manifolds, in particular. These results will be obtained by using theorems of the connections between the geometry of a complete smooth manifold and the global behavior of its subharmonic functions. In conclusion, we show applications of this method for global geometry of a complete locally conformally flat Riemannian manifold with constant scalar curvature because its Ricci tensor is a Codazzi tensor and for global geometry of a complete hypersurface in a standard sphere because its second fundamental form is also a Codazzi tensor.

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On conformally flat manifolds with constant positive scalar curvature

Cited by 26 publications

References 27 publications

Four-manifolds with positive Yamabe constant

Four-manifolds with positive Yamabe constant

A potential generalization of some canonical Riemannian metrics

On higher-order Codazzi tensors on complete Riemannian manifolds

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