• Title/Summary/Keyword: Ricci curvature

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ON LIGHTLIKE HYPERSURFACES OF A GRW SPACE-TIME

  • Kang, Tae-Ho
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.4
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    • pp.863-874
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    • 2012
  • We provide a study of lightlike hypersurfaces of a generalized Robertson-Walker (GRW) space-time. In particular, we investigate lightlike hypersurfaces with curvature invariance, parallel second fundamental forms, totally umbilical second fundamental forms, null sectional curvatures and null Ricci curvatures, respectively.

ON CURVATURE PINCHING FOR TOTALLY REAL SUBMANIFOLDS OF $H^n$(c)

  • Matsuyama, Yoshio
    • Journal of the Korean Mathematical Society
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    • v.34 no.2
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    • pp.321-336
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    • 1997
  • Let S be the Ricci curvature of an n-dimensional compact minimal totally real submanifold M of a quaternion projective space $HP^n (c)$ of quaternion sectional curvature c. We proved that if $S \leq \frac{16}{3(n -2)}c$, then either $S \equiv \frac{4}{n - 1}c$ (i.e. M is totally geodesic or $S \equiv \frac{16}{3(n - 2)}c$. All compact minimal totally real submanifolds of $HP^n (c)$ satisfy in $S \equiv \frac{16}{3(n - 2)}c$ are determined.

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THE EXPANSION OF MEAN DISTANCE OF BROWNIAN MOTION ON RIEMANNIAN MANIFOLD

  • Kim, Yoon-Tae;Park, Hyun-Suk;Jeon, Jong-Woo
    • Proceedings of the Korean Statistical Society Conference
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    • 2003.05a
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    • pp.37-42
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    • 2003
  • We study the asymptotic expansion in small time of the mean distance of Brownian motion on Riemannian manifolds. We compute the first four terms of the asymptotic expansion of the mean distance by using the decomposition of Laplacian into homogeneous components. This expansion can he expressed in terms of the scalar valued curvature invariants of order 2, 4, 6.

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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.

ALGEBRAIC RICCI SOLITONS IN THE FINSLERIAN CASE

  • Jiao, Guocheng;Yan, Zaili
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.4
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    • pp.853-863
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    • 2021
  • In this paper, we study algebraic Ricci solitons in the Finslerian case. We show that any simply connected Finslerian algebraic Ricci soliton is a Finslerian Ricci soliton. Furthermore, we study Randers algebraic Ricci solitons. It turns out that a shrinking, steady, or expanding Randers algebraic Ricci soliton with vanishing S-curvature is Einstein, locally Minkowskian, or Riemannian, respectively.

On N(κ)-Contact Metric Manifolds Satisfying Certain Curvature Conditions

  • De, Avik;Jun, Jae-Bok
    • Kyungpook Mathematical Journal
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    • v.51 no.4
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    • pp.457-468
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    • 2011
  • We consider pseudo-symmetric and Ricci generalized pseudo-symmetric N(${\kappa}$) contact metric manifolds. We also consider N(${\kappa}$)-contact metric manifolds satisfying the condition $S{\cdot}R$ = 0 where R and S denote the curvature tensor and the Ricci tensor respectively. Finally we give some examples.

CONFORMAL RICCI SOLITON ON PARACONTACT METRIC (k, 𝜇)-MANIFOLDS WITH SCHOUTEN-VAN KAMPEN CONNECTION

  • Pardip Mandal;Mohammad Hasan Shahid;Sarvesh Kumar Yadav
    • Communications of the Korean Mathematical Society
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    • v.39 no.1
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    • pp.161-173
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    • 2024
  • The main object of the present paper is to study conformal Ricci soliton on paracontact metric (k, 𝜇)-manifolds with respect to Schouten-van Kampen connection. Further, we obtain the result when paracontact metric (k, 𝜇)-manifolds with respect to Schouten-van Kampen connection satisfying the condition $^*_C({\xi},U){\cdot}^*_S=0$. Finally we characterized concircular curvature tensor on paracontact metric (k, 𝜇)-manifolds with respect to Schouten-van Kampen connection.

ON A CLASSIFICATION OF WARPED PRODUCT SPACES WITH GRADIENT RICCI SOLITONS

  • Lee, Sang Deok;Kim, Byung Hak;Choi, Jin Hyuk
    • Korean Journal of Mathematics
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    • v.24 no.4
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    • pp.627-636
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    • 2016
  • In this paper, we study Ricci solitons, gradient Ricci solitons in the warped product spaces and gradient Yamabe solitons in the Riemannian product spaces. We obtain the necessary and sufficient conditions for the Riemannian product spaces to be Ricci solitons. Moreover we classify the warped product space which admit gradient Ricci solitons under some conditions of the potential function.

From the Eisenhart Problem to Ricci Solitons in Quaternion Space Forms

  • Praveena, Mundalamane Manjappa;Bagewadi, Channabasappa Shanthappa
    • Kyungpook Mathematical Journal
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    • v.58 no.2
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    • pp.389-398
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    • 2018
  • In this paper we obtain the condition for the existence of Ricci solitons in nonflat quaternion space form by using Eisenhart problem. Also it is proved that if (g, V, ${\lambda}$) is Ricci soliton then V is solenoidal if and only if it is shrinking, steady and expanding depending upon the sign of scalar curvature. Further it is shown that Ricci soliton in semi-symmetric quaternion space form depends on quaternion sectional curvature c if V is solenoidal.

STUDY OF GRADIENT SOLITONS IN THREE DIMENSIONAL RIEMANNIAN MANIFOLDS

  • Biswas, Gour Gopal;De, Uday Chand
    • Communications of the Korean Mathematical Society
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    • v.37 no.3
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    • pp.825-837
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    • 2022
  • We characterize a three-dimensional Riemannian manifold endowed with a type of semi-symmetric metric P-connection. At first, it is proven that if the metric of such a manifold is a gradient m-quasi-Einstein metric, then either the gradient of the potential function 𝜓 is collinear with the vector field P or, λ = -(m + 2) and the manifold is of constant sectional curvature -1, provided P𝜓 ≠ m. Next, it is shown that if the metric of the manifold under consideration is a gradient 𝜌-Einstein soliton, then the gradient of the potential function is collinear with the vector field P. Also, we prove that if the metric of a 3-dimensional manifold with semi-symmetric metric P-connection is a gradient 𝜔-Ricci soliton, then the manifold is of constant sectional curvature -1 and λ + 𝜇 = -2. Finally, we consider an example to verify our results.