Ricci solitons and real hypersurfaces in a complex space form

Cho, Jong Taek; Kimura, Makoto

doi:10.2748/tmj/1245849443

Cited by 192 publications

(155 citation statements)

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“…Ricci solitons on Riemannian submanifolds have also been studied in [10,11,12] by J. T. Cho and M. Kimura from a different viewpoint. They proved several interesting results on Ricci solitons on submanifolds; however their potential fields of the Ricci solitons are quite different from ours.…”

Section: Ricci Solitons On Riemannian Submanifolds Arisen From Concirmentioning

confidence: 99%

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Some Results on Concircular Vector Fields and Their Applications to Ricci Solitons

Chen¹

2015

Bulletin of the Korean Mathematical Society

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Abstract. A vector field on a Riemannian manifold (M, g) is called concircular if it satisfies ∇ X v = µX for any vector X tangent to M , where ∇ is the Levi-Civita connection and µ is a non-trivial function on M . A smooth vector field ξ on a Riemannian manifold (M, g) is said to define a Ricci soliton if it satisfies the following Ricci soliton equation:where L ξ g is the Lie-derivative of the metric tensor g with respect to ξ, Ric is the Ricci tensor of (M, g) and λ is a constant. A Ricci soliton (M, g, ξ, λ) on a Riemannian manifold (M, g) is said to have concircular potential field if its potential field ξ is a concircular vector field.In the first part of this paper we determine Riemannian manifolds which admit a concircular vector field. In the second part we classify Ricci solitons with concircular potential field. In the last part we prove some important properties of Ricci solitons on submanifolds of a Riemannian manifold equipped with a concircular vector field.

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Section: Ricci Solitons On Riemannian Submanifolds Arisen From Concirmentioning

confidence: 99%

“…A Ricci soliton (M, g, ξ, λ) is called shrinking, steady or expanding according to λ > 0, λ = 0, or λ < 0, respectively. A trivial Ricci soliton is one for which ξ is zero or Killing, in which case the metric is Einsteinian (see, for instance, [6,7,10,14,16,17]). Compact Ricci solitons are the fixed points of the Ricci flow:…”

Section: Introductionmentioning

confidence: 99%

Some Results on Concircular Vector Fields and Their Applications to Ricci Solitons

Chen¹

2015

Bulletin of the Korean Mathematical Society

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“…Cho and Kimura [3] studied on Ricci solitons of real hypersurfaces in a nonflat complex space form and showed that a real hypersurface M in a non-flat complex space form M n (c = 0) does not admit a Ricci soliton such that the Reeb vector field ξ is potential vector field. They defined so called η-Ricci soliton, such that satisfies…”

Section: Introductionmentioning

confidence: 99%

Ricci solitons on QR-hypersurfaces of a quaternionic space form ℚⁿ

Nazari

Abedi

2016

Acta Universitatis Sapientiae, Mathematica

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“…In [6], [2] and [9], the authors proved that there are no Einstein real hypersurfaces of non- ‡at complex space forms. Motivated by this the authors Cho and Kimura [3] introduced the notion of -Ricci solitons and gave a classi…cation of real hypersurfaces in non- ‡at complex space forms admitting -Ricci solitons. Later Blaga [1] studied -Ricci solitons in paraKenmotsu manifolds.…”

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confidence: 99%