Conformal transformations in complete product Riemannian manifolds

Tashiro, Yoshihiro; Miyashita, Koichi

doi:10.2969/jmsj/01930328

Cited by 20 publications

(6 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…For the case of a 2-dimensional base see [1,Thm.9.119]. The similar equations ∇ 2 f = − kf g and ∇ 2 * f = kf g * hold for the divergence f of a non-isometric conformal vector field on a complete non-flat Riemannian product M × M * , see [38,Thm.5], [42]. The crucial condition k = −k * occurs also in the Fefferman-Graham ambient metric construction on M × M * × [0, ε), see [16].…”

Section: Remarkmentioning

confidence: 99%

Conformally Einstein product spaces

Kühnel

Rademacher

2016

Differential Geometry and its Applications

View full text Add to dashboard Cite

We study pseudo-Riemannian Einstein manifolds which are conformally equivalent with a metric product of two pseudo-Riemannian manifolds. Particularly interesting is the case where one of these manifolds is 1-dimensional and the case where the conformal factor depends on both manifolds simultaneously. If both factors are at least 3-dimensional then the latter case reduces to the product of two Einstein spaces, each of the special type admitting a non-trivial conformal gradient field. These are completely classified. If each factor is 2-dimensional, there is a special family of examples of non-constant curvature (called extremal metrics by Calabi), where in each factor the gradient of the Gaussian curvature is a conformal vector field. Then the metric of the 2-manifold is a warped product where the warping function is the first derivative of the Gaussian curvature. Moreover we find explicit examples of Einstein warped products with a 1-dimensional fibre and such with a 2-dimensional base. Therefore in the 4dimensional case our Main Theorem points towards a local classification of conformally Einstein products. Finally we prove an assertion in the book by A.Besse on complete Einstein warped products with a 2-dimensional base. All solutions can be explicitly written in terms of integrals of elementary functions.

show abstract

Section: Remarkmentioning

confidence: 99%

Conformally Einstein product spaces

Kühnel

Rademacher

2016

Differential Geometry and its Applications

View full text Add to dashboard Cite

show abstract

“…Then we have Δiaψ = Δ,ia)f = aΔf = aλf = aλf* , and therefore J (<*)/* = «/* for a = n/λ. But such a product manifold M(a) cannot admit a non-isometric conformal vector field [11], [14]. This shows that a condition is needed for a function / with Δf -nkf to be the gradient of a conformal vector field on a Riemannian manifold with constant scalar curvature k. However, only a sufficient condition is known for compact M in the following theorem [9], [10]:…”

Section: Final Miscellanymentioning

confidence: 99%

Conformal transformations of Riemannian manifolds

Obata¹

1970

J. Differential Geom.

View full text Add to dashboard Cite

“…The proof of this theorem depends on the following lemma. For a related work, see [9,Proposition 7.3]. Lemma 3.3.…”

Section: The Proof Of Theorem Bmentioning

confidence: 99%

Riemannian Invariants that Characterize Rotational Symmetries of the Standard Sphere

Aino

2016

Preprint

View full text Add to dashboard Cite

Inspired by the Lichnerowicz-Obata theorem for the first eigenvalue of the Laplacian, we define a new family of invariants {Ω k (g)} for closed Riemannian manifolds. The value of Ω k (g) sharply reflects the spherical part of the manifold. Indeed, Ω 1 (g) and Ω 2 (g) characterize the standard sphere. Contents 16 4. Computation and examples 18 4.1. The product of Einstein manifolds 18 4.2. The case of Heisenberg manifolds 19 Appendix A. The proof of Lemma 2.5 24 References 26

show abstract

Conformal transformations in complete product Riemannian manifolds

Cited by 20 publications

References 7 publications

Conformally Einstein product spaces

Conformally Einstein product spaces

Conformal transformations of Riemannian manifolds

Riemannian Invariants that Characterize Rotational Symmetries of the Standard Sphere

Contact Info

Product

Resources

About