• 제목/요약/키워드: scalar curvature

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A NOTE ON DECREASING SCALAR CURVATURE FROM FLAT METRICS

  • Kim, Jongsu
    • 호남수학학술지
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    • 제35권4호
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    • pp.647-655
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    • 2013
  • We obtain $C^{\infty}$-continuous paths of explicit Riemannian metrics $g_t$, $0{\leq}t$ < ${\varepsilon}$, whose scalar curvatures $s(g_t)$ decrease, where $g_0$ is a flat metric, i.e. a metric with vanishing curvature. Most of them can exist on tori of dimension ${\geq}3$. Some of them yield scalar curvature decrease on a ball in the Euclidean space.

ZERO SCALAR CURVATURE ON OPEN MANIFOLDS

  • Kim, Seong-Tag
    • 대한수학회논문집
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    • 제13권3호
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    • pp.539-544
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    • 1998
  • Let (M, g) be a noncompact complete Riemannian manifold of dimension n $\geq$ 3 with scalar curvature S, which is close to O. With conditions on a conformal invariant and scalar curvature of (M, g), we show that there exists a conformal metric (equation omitted), near g, whose scalar curvature (equation omitted) = 0 by gluing solutions of the corresponding partial differential equation on each bounded subsets $K_{i}$ with ∪$K_{i}$ = M.

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GRADIENT YAMABE SOLITONS WITH CONFORMAL VECTOR FIELD

  • Fasihi-Ramandi, Ghodratallah;Ghahremani-Gol, Hajar
    • 대한수학회논문집
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    • 제36권1호
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    • pp.165-171
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    • 2021
  • The purpose of this paper is to investigate the geometry of complete gradient Yamabe soliton (Mn, g, f, λ) with constant scalar curvature admitting a non-homothetic conformal vector field V leaving the potential vector field invariant. We show that in such manifolds the potential function f is constant and the scalar curvature of g is determined by its soliton scalar. Considering the locally conformally flat case and conformal vector field V, without constant scalar curvature assumption, we show that g has constant curvature and determines the potential function f explicitly.

RICCI AND SCALAR CURVATURES ON SU(3)

  • Kim, Hyun-Woong;Pyo, Yong-Soo;Shin, Hyun-Ju
    • 호남수학학술지
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    • 제34권2호
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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.

DEFORMATION OF CARTAN CURVATURE ON FINSLER MANIFOLDS

  • Bidabad, Behroz;Shahi, Alireza;Ahmadi, Mohamad Yar
    • 대한수학회보
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    • 제54권6호
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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.

TOTAL SCALAR CURVATURE AND EXISTENCE OF STABLE MINIMAL SURFACES

  • Hwang, Seung-Su
    • 호남수학학술지
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    • 제30권4호
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    • pp.677-683
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    • 2008
  • On a compact n-dimensional manifold M, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of volume 1, should be Einstein. The purpose of the present paper is to prove that a 3-dimensional manifold (M,g) is isometric to a standard sphere if ker $s^*_g{{\neq}}0$ and there is a lower Ricci curvature bound. We also study the structure of a compact oriented stable minimal surface in M.

DIFFERENTIAL EQUATIONS ON WARPED PRODUCTS (II)

  • JUNG, YOON-TAE
    • 호남수학학술지
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    • 제28권3호
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    • pp.399-407
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    • 2006
  • In this paper, we consider the problem of achieving a prescribed scalar curvature on warped product manifolds according to fiber manifolds with zero scalar curvature.

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WEAKLY EINSTEIN CRITICAL POINT EQUATION

  • Hwang, Seungsu;Yun, Gabjin
    • 대한수학회보
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    • 제53권4호
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    • pp.1087-1094
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    • 2016
  • On a compact n-dimensional manifold M, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, after derivng an interesting curvature identity, we show that the conjecture is true in dimension three and four when g is weakly Einstein. In higher dimensional case $n{\geq}5$, we also show that the conjecture is true under an additional Ricci curvature bound. Moreover, we prove that the manifold is isometric to a standard n-sphere when it is n-dimensional weakly Einstein and the kernel of the linearized scalar curvature operator is nontrivial.

GENERALIZED LANDSBERG MANIFOLDS OF SCALAR CURVATURE

  • Aurel Bejancu;Farran, Hani-Reda
    • 대한수학회보
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    • 제37권3호
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    • pp.543-550
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    • 2000
  • We prove that every generalized Landsberg manifold of scalar curvature R is a Riemannian manifold of constant curvature, provided that $R\neq\ 0$.

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