• 제목/요약/키워드: k-Ricci curvature

검색결과 119건 처리시간 0.018초

ON RICCI CURVATURES OF LEFT INVARIANT METRICS ON SU(2)

  • Pyo, Yong-Soo;Kim, Hyun-Woong;Park, Joon-Sik
    • 대한수학회보
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    • 제46권2호
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    • pp.255-261
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    • 2009
  • In this paper, we shall prove several results concerning Ricci curvature of a Riemannian manifold (M, g) := (SU(2), g) with an arbitrary given left invariant metric g. First of all, we obtain the maximum (resp. minimum) of {r(X) := Ric(X,X) | ${||X||}_g$ = 1,X ${\in}$ X(M)}, where Ric is the Ricci tensor field on (M, g), and then get a necessary and sufficient condition for the Levi-Civita connection ${\nabla}$ on the manifold (M, g) to be projectively flat. Furthermore, we obtain a necessary and sufficient condition for the Ricci curvature r(X) to be always positive (resp. negative), independently of the choice of unit vector field X.

COMPARISON THEOREMS IN FINSLER GEOMETRY WITH WEIGHTED CURVATURE BOUNDS AND RELATED RESULTS

  • Wu, Bing-Ye
    • 대한수학회지
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    • 제52권3호
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    • pp.603-624
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    • 2015
  • We first extend the notions of weighted curvatures, including the weighted flag curvature and the weighted Ricci curvature, for a Finsler manifold with given volume form. Then we establish some basic comparison theorems for Finsler manifolds with various weighted curvature bounds. As applications, we obtain some McKean type theorems for the first eigenvalue of Finsler manifolds, some results on weighted curvature and fundamental group for Finsler manifolds, as well as an estimation of Gromov simplicial norms for reversible Finsler manifolds.

MANIFOLDS WITH NONNEGATIVE RICCI CURVATURE ALMOST EVERYWHERE

  • Paeng, Seong-Hun
    • 대한수학회지
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    • 제36권1호
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    • pp.125-137
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    • 1999
  • Under the condition of RicM $\geq$ -(n-1), injM $\geq$ I0, we prove the existence of an $\varepsilon$>0 such that on the region of volume $\varepsilon$>0 the curvature condition of splitting theorem can be weakened.

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STATIC AND RELATED CRITICAL SPACES WITH HARMONIC CURVATURE AND THREE RICCI EIGENVALUES

  • Kim, Jongsu
    • 대한수학회지
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    • 제57권6호
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    • pp.1435-1449
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    • 2020
  • In this article we make a local classification of n-dimensional Riemannian manifolds (M, g) with harmonic curvature and less than four Ricci eigenvalues which admit a smooth non constant solution f to the following equation $$(1)\hspace{20}{\nabla}df=f(r-{\frac{R}{n-1}}g)+x{\cdot} r+y(R)g,$$ where ∇ is the Levi-Civita connection of g, r is the Ricci tensor of g, x is a constant and y(R) a function of the scalar curvature R. Indeed, we showed that, in a neighborhood V of each point in some open dense subset of M, either (i) or (ii) below holds; (i) (V, g, f + x) is a static space and isometric to a domain in the Riemannian product of an Einstein manifold N and a static space (W, gW, f + x), where gW is a warped product metric of an interval and an Einstein manifold. (ii) (V, g) is isometric to a domain in the warped product of an interval and an Einstein manifold. For the proof we use eigenvalue analysis based on the Codazzi tensor properties of the Ricci tensor.

THE EXPANSION OF MEAN DISTANCE OF BROWNIAN MOTION ON RIEMANNIAN MANIFOLD

  • Kim, Yoon-Tae;Park, Hyun-Suk;Jeon, Jong-Woo
    • 한국통계학회:학술대회논문집
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    • 한국통계학회 2003년도 춘계 학술발표회 논문집
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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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CONFORMAL RICCI SOLITON ON PARACONTACT METRIC (k, 𝜇)-MANIFOLDS WITH SCHOUTEN-VAN KAMPEN CONNECTION

  • Pardip Mandal;Mohammad Hasan Shahid;Sarvesh Kumar Yadav
    • 대한수학회논문집
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    • 제39권1호
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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.

CRITICAL POINT METRICS OF THE TOTAL SCALAR CURVATURE

  • Chang, Jeong-Wook;Hwang, Seung-Su;Yun, Gab-Jin
    • 대한수학회보
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    • 제49권3호
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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.

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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    • 제24권4호
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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.

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

  • De, Avik;Jun, Jae-Bok
    • Kyungpook Mathematical Journal
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    • 제51권4호
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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.