• Title, Summary, Keyword: Ricci curvature

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RICCI SOLITONS ON RICCI PSEUDOSYMMETRIC (LCS)n-MANIFOLDS

  • Hui, Shyamal Kumar;Lemence, Richard S.;Chakraborty, Debabrata
    • Honam Mathematical Journal
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    • v.40 no.2
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    • pp.325-346
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    • 2018
  • The object of the present paper is to study some types of Ricci pseudosymmetric $(LCS)_n$-manifolds whose metric is Ricci soliton. We found the conditions when Ricci soliton on concircular Ricci pseudosymmetric, projective Ricci pseudosymmetric, $W_3$-Ricci pseudosymmetric, conharmonic Ricci pseudosymmetric, conformal Ricci pseudosymmetric $(LCS)_n$-manifolds to be shrinking, steady and expanding. We also construct an example of concircular Ricci pseudosymmetric $(LCS)_3$-manifold whose metric is Ricci soliton.

THE RICCI CURVATURE ON DIRECTED GRAPHS

  • Yamada, Taiki
    • Journal of the Korean Mathematical Society
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    • v.56 no.1
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    • pp.113-125
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    • 2019
  • In this paper, we consider the Ricci curvature of a directed graph, based on Lin-Lu-Yau's definition. We give some properties of the Ricci curvature, including conditions for a directed regular graph to be Ricci-flat. Moreover, we calculate the Ricci curvature of the cartesian product of directed graphs.

A NEW 3-PARAMETER CURVATURE CONDITION PRESERVED BY RICCI FLOW

  • Gao, Xiang
    • Journal of the Korean Mathematical Society
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    • v.50 no.4
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    • pp.829-845
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    • 2013
  • In this paper, we firstly establish a family of curvature invariant conditions lying between the well-known 2-nonnegative curvature operator and nonnegative curvature operator along the Ricci flow. These conditions are defined by a set of inequalities involving the first four eigenvalues of the curvature operator, which are named as 3-parameter ${\lambda}$-nonnegative curvature conditions. Then a related rigidity property of manifolds with 3-parameter ${\lambda}$-nonnegative curvature operators is also derived. Based on these, we also obtain a strong maximum principle for the 3-parameter ${\lambda}$-nonnegativity along Ricci flow.

GENERALIZED MYERS THEOREM FOR FINSLER MANIFOLDS WITH INTEGRAL RICCI CURVATURE BOUND

  • Wu, Bing-Ye
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.4
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    • pp.841-852
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    • 2019
  • We establish the generalized Myers theorem for Finsler manifolds under integral Ricci curvature bound. More precisely, we show that the forward complete Finsler n-manifold whose part of Ricci curvature less than a positive constant is small in $L^p$-norm (for p > n/2) have bounded diameter and finite fundamental group.

RICCI CURVATURE OF SUBMANIFOLDS IN A QUATERNION PROJECTIVE SPACE

  • Liu, Ximin;Dai, Wanji
    • Communications of the Korean Mathematical Society
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    • v.17 no.4
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    • pp.625-633
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    • 2002
  • Recently, Chen establishes sharp relationship between the k-Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. In this paper, we establish sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in quaternion projective spaces.

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.

RICCI CURVATURE, CIRCULANTS, AND EXTENDED MATCHING CONDITIONS

  • Dagli, Mehmet;Olmez, Oktay;Smith, Jonathan D.H.
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.1
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    • pp.201-217
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    • 2019
  • Ricci curvature for locally finite graphs, as proposed by Lin, Lu and Yau, provides a useful isomorphism invariant. A Matching Condition was introduced as a key tool for computation of this Ricci curvature. The scope of the Matching Condition is quite broad, but it does not cover all cases. Thus the current paper introduces extended versions of the Matching Condition, and applies them to the computation of the Ricci curvature of a class of circulants determined by certain number-theoretic data. The classical Matching Condition is also applied to determine the Ricci curvature for other families of circulants, along with Cayley graphs of abelian groups that are generated by the complements of (unions of) subgroups.

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.

SOME INEQUALITIES ON TOTALLY REAL SUBMANIFOLDS IN LOCALLY CONFORMAL KAEHLER SPACE FORMS

  • Alfonso, Carriazo;Kim, Young-Ho;Yoon, Dae-Won
    • Journal of the Korean Mathematical Society
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    • v.41 no.5
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    • pp.795-808
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    • 2004
  • In this article, we establish sharp relations between the sectional curvature and the shape operator and also between the k-Ricci curvature and the shape operator for a totally real submanifold in a locally conformal Kaehler space form of constant holomorphic sectional curvature with arbitrary codimension. mean curvature, sectional curvature, shape operator, k-Ricci curvature, locally conformal Kaehler space form, totally real submanifold.