• Title/Summary/Keyword: Ricci scalar

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ON EVOLUTION OF FINSLER RICCI SCALAR

  • Bidabad, Behroz;Sedaghat, Maral Khadem
    • Journal of the Korean Mathematical Society
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    • v.55 no.3
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    • pp.749-761
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    • 2018
  • Here, we calculate the evolution equation of the reduced hh-curvature and the Ricci scalar along the Finslerian Ricci flow. We prove that Finsler Ricci flow preserves positivity of the reduced hh-curvature on finite time. Next, it is shown that evolution of Ricci scalar is a parabolic-type equation and moreover if the initial Finsler metric is of positive flag curvature, then the flag curvature, as well as the Ricci scalar, remain positive as long as the solution exists. Finally, we present a lower bound for Ricci scalar along Ricci flow.

RICCI AND SCALAR CURVATURES ON SU(3)

  • Kim, Hyun-Woong;Pyo, Yong-Soo;Shin, Hyun-Ju
    • Honam Mathematical Journal
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    • v.34 no.2
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    • pp.231-239
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    • 2012
  • In this paper, we obtain the Ricci curvature and the scalar curvature on SU(3) with some left invariant Riemannian metric. And then we get a necessary and sufficient condition for the scalar curvature (resp. the Ricci curvature) on the Riemannian manifold SU(3) to be positive.

ON QUASI RICCI SYMMETRIC MANIFOLDS

  • Kim, Jaeman
    • Korean Journal of Mathematics
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    • v.27 no.1
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    • pp.9-15
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    • 2019
  • In this paper, we study a type of Riemannian manifold, namely quasi Ricci symmetric manifold. Among others, we show that the scalar curvature of a quasi Ricci symmetric manifold is constant. In addition if the manifold is Einstein, then its Ricci tensor is zero. Also we prove that if the associated vector field of a quasi Ricci symmetric manifold is either recurrent or concurrent, then its Ricci tensor is zero.

On Conformally at Almost Pseudo Ricci Symmetric Mani-folds

  • De, Uday Chand;Gazi, Abul Kalam
    • Kyungpook Mathematical Journal
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    • v.49 no.3
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    • pp.507-520
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    • 2009
  • The object of the present paper is to study conformally at almost pseudo Ricci symmetric manifolds. The existence of a conformally at almost pseudo Ricci symmetric manifold with non-zero and non-constant scalar curvature is shown by a non-trivial example. We also show the existence of an n-dimensional non-conformally at almost pseudo Ricci symmetric manifold with vanishing scalar curvature.

DEFORMATION OF CARTAN CURVATURE ON FINSLER MANIFOLDS

  • Bidabad, Behroz;Shahi, Alireza;Ahmadi, Mohamad Yar
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.6
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    • pp.2119-2139
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    • 2017
  • Here, certain Ricci flow for Finsler n-manifolds is considered and deformation of Cartan hh-curvature, as well as Ricci tensor and scalar curvature, are derived for spaces of scalar flag curvature. As an application, it is shown that on a family of Finsler manifolds of constant flag curvature, the scalar curvature satisfies the so-called heat-type equation. Hence on a compact Finsler manifold of constant flag curvature of initial non-negative scalar curvature, the scalar curvature remains non-negative by Ricci flow and blows up in a short time.

PARA-KENMOTSU METRIC AS A 𝜂-RICCI SOLITON

  • Kundu, Satyabrota
    • Korean Journal of Mathematics
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    • v.29 no.2
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    • pp.445-453
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    • 2021
  • The purpose of the paper is to study of Para-Kenmotsu metric as a 𝜂-Ricci soliton. The paper is organized as follows: • If an 𝜂-Einstein para-Kenmotsu metric represents an 𝜂-Ricci soliton with flow vector field V, then it is Einstein with constant scalar curvature r = -2n(2n + 1). • If a para-Kenmotsu metric g represents an 𝜂-Ricci soliton with the flow vector field V being an infinitesimal paracontact transformation, then V is strict and the manifold is an Einstein manifold with constant scalar curvature r = -2n(2n + 1). • If a para-Kenmotsu metric g represents an 𝜂-Ricci soliton with non-zero flow vector field V being collinear with 𝜉, then the manifold is an Einstein manifold with constant scalar curvature r = -2n(2n + 1). Finally, we cited few examples to illustrate the results obtained.

CRITICAL POINT METRICS OF THE TOTAL SCALAR CURVATURE

  • Chang, Jeong-Wook;Hwang, Seung-Su;Yun, Gab-Jin
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.3
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    • pp.655-667
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    • 2012
  • In this paper, we deal with a critical point metric of the total scalar curvature on a compact manifold $M$. We prove that if the critical point metric has parallel Ricci tensor, then the manifold is isometric to a standard sphere. Moreover, we show that if an $n$-dimensional Riemannian manifold is a warped product, or has harmonic curvature with non-parallel Ricci tensor, then it cannot be a critical point metric.

CURVATURE ESTIMATES FOR GRADIENT EXPANDING RICCI SOLITONS

  • Zhang, Liangdi
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.3
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    • pp.537-557
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    • 2021
  • In this paper, we investigate the curvature behavior of complete noncompact gradient expanding Ricci solitons with nonnegative Ricci curvature. For such a soliton in dimension four, it is shown that the Riemann curvature tensor and its covariant derivatives are bounded. Moreover, the Ricci curvature is controlled by the scalar curvature. In higher dimensions, we prove that the Riemann curvature tensor grows at most polynomially in the distance function.

On a Normal Contact Metric Manifold

  • Calin, Constantin;Ispas, Mihai
    • Kyungpook Mathematical Journal
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    • v.45 no.1
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    • pp.55-65
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    • 2005
  • We find the expression of the curvature tensor field for a manifold with is endowed with an almost contact structure satisfying the condition (1.7). By using this condition we obtain some properties of the Ricci tensor and scalar curvature (d. Theorem 3.2 and Proposition 3.2).

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