On products of harmonic forms

Kotschick, D.

doi:10.1215/s0012-7094-01-10734-5

Cited by 60 publications

(116 citation statements)

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“…First, symplectic pairs appear naturally in the study of Riemannian metrics for which all products of harmonic forms are harmonic (see [16]) and in the investigation of the group cohomology of symplectomorphism groups; see [17]. Second, symplectic pairs can be used to construct new examples of contactsymplectic and of contact pairs in the sense of [1,2,4].…”

Section: Introductionmentioning

confidence: 99%

The geometry of symplectic pairs

Bande

Kotschick

2005

Trans. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

The geometry of symplectic pairs

Bande

Kotschick

2005

Trans. Amer. Math. Soc.

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“…We will show that a warped product metric on a compact manifold is formal if and only if the warping function is constant. On the way, we shall also provide a proof for the fact (stated in [Kotschick 2001], for instance) that a product of formal metrics is formal.…”

Section: Introductionmentioning

confidence: 91%

“…This motivated the following definition: Definition 1.1 [Kotschick 2001]. A closed manifold is called geometrically formal if it admits a formal Riemannian metric.…”

Section: Introductionmentioning

confidence: 99%

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Remarks on the product of harmonic forms

Pilca¹

2011

Pacific J. Math.

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A metric is formal if all products of harmonic forms are again harmonic. The existence of a formal metric implies Sullivan formality of the manifold, and hence formal metrics can exist only in the presence of a very restricted topology. We show that a warped product metric is formal if and only if the warping function is constant and derive further topological obstructions to the existence of formal metrics. In particular, we determine the necessary and sufficient conditions for a Vaisman metric to be formal. IntroductionA fundamental problem in algebraic topology is the reading of the homotopy type of a space in terms of cohomological data. A precise definition of this property was given by Sullivan [1977] and called formality. As concerns manifolds, it is known, for example, that all compact Riemannian symmetric spaces and all compact Kähler manifolds are formal. For a recent survey of topological formality, see [Papadima and Suciu 2009].Sullivan also observed that if a compact manifold admits a metric such that the wedge product of any two harmonic forms is again harmonic, then, by Hodge theory, the manifold is formal. This motivated the following definition: In particular, the length of any harmonic form with respect to a formal metric is (pointwise) constant. This larger class of metrics having all harmonic (one-)forms of constant length naturally appears in other geometric contexts, for instance in the study of certain systolic inequalities, and has been investigated in [Nagy 2006;Nagy and Vernicos 2004].Classical examples of geometrically formal manifolds are compact symmetric spaces. In [Kotschick and Terzić 2003;2011] more general examples are provided, Both authors are partially supported by CNCSIS grant PNII IDEI contract 529/2009. M. Pilca also acknowledges partial support from SFB/TR 12. MSC2000: primary 53C25; secondary 53C55, 58A14.

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“…These are closed oriented Riemannian manifolds such that the product of two harmonic forms is still harmonic. They have the property that all harmonic forms have constant pointwise norm (see [Kot01]). In particular their first Betti number cannot be one less than the dimension and if it equals the dimension, then it is a flat torus.…”

Section: Introductionmentioning

confidence: 99%

The length of harmonic forms on a compact Riemannian manifold

Nagy¹,

Vernicos²

2004

Trans. Amer. Math. Soc.

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Abstract. We study (n + 1)-dimensional Riemannian manifolds with harmonic forms of constant length and first Betti number equal to n showing that they are 2-step nilmanifolds with some special metrics. We also characterize, in terms of properties on the product of harmonic forms, the left-invariant metrics among them. This allows us to clarify the case of equality in the stable isosytolic inequalities in that setting. We also discuss other values of the Betti number.

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On products of harmonic forms

Cited by 60 publications

References 6 publications

The geometry of symplectic pairs

The geometry of symplectic pairs

Remarks on the product of harmonic forms

The length of harmonic forms on a compact Riemannian manifold

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