• Title, Summary, Keyword: projectively flat

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ON A CLASS OF LOCALLY PROJECTIVELY FLAT GENERAL (α, β)-METRICS

  • Mo, Xiaohuan;Zhu, Hongmei
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.4
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    • pp.1293-1307
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    • 2017
  • General (${\alpha},{\beta}$)-metrics form a rich class of Finsler metrics. They include many important Finsler metrics, such as Randers metrics, square metrics and spherically symmetric metrics. In this paper, we find equations which are necessary and sufficient conditions for such Finsler metric to be locally projectively flat. By solving these equations, we obtain all of locally projectively flat general (${\alpha},{\beta}$)-metrics under certain condition. Finally, we manufacture explicitly new locally projectively flat Finsler metrics.

LIMIT SETS OF PROJECTIVELY FLAT MANIFOLDS

  • Park, Kyeong-Su
    • Communications of the Korean Mathematical Society
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    • v.15 no.3
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    • pp.541-547
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    • 2000
  • In this paper, we discuss various limit sets of projectively flat manifolds and relationship between them.

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WEYL STRUCTURES ON COMPACT CONNECTED LIE GROUPS

  • Park, Joon-Sik;Pyo, Yong-Soo;Shin, Young-Lim
    • Journal of the Chungcheong Mathematical Society
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    • v.24 no.3
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    • pp.503-515
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    • 2011
  • Let G be a compact connected semisimple Lie group, B the Killing form of the algebra g of G, and g the invariant metric induced by B. Then, we obtain a necessary and sufficient condition for a left invariant linear connection D with a Weyl structure ($D,\;g,\;{\omega}$) on (G, g) to be projectively flat (resp. Einstein-Weyl). And, we also get that if a left invariant linear connection D with a Weyl structure ($D,\;g,\;{\omega}$) on (G, g) which has symmetric Ricci tensor $Ric^D$ is projectively flat, then the connection D is Einstein-Weyl; but the converse is not true. Moreover, we show that if a left invariant connection D with Weyl structure ($D,\;g,\;{\omega}$) on (G, g) is projectively flat (resp. Einstein-Weyl), then D is a Yang-Mills connection.

ON PROJECTIVELY FLAT FINSLER SPACES WITH $({\alpha},{\beta})$-METRIC

  • Park, Hong-Suh;Lee, Il-Yong
    • Communications of the Korean Mathematical Society
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    • v.14 no.2
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    • pp.373-383
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    • 1999
  • The ($\alpha$,$\beta$)-metric is a Finsler metric which is constructed from a Riemannian metric $\alpha$ and a differential 1-from $\beta$;it has been sometimes treated in theoretical physics. The condition for a Finsler space with an ($\alpha$,$\beta$)-metric L($\alpha$,$\beta$) to be projectively flat was given by Matsumoto [11]. The present paper is devoted to studying the condition for a Finsler space with L=$\alpha$\ulcorner$\beta$\ulcorner or L=$\alpha$+$\beta$\ulcorner/$\alpha$ to be projectively flat on the basis of Matsumoto`s results.

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SOME CLASSES OF 3-DIMENSIONAL NORMAL ALMOST PARACONTACT METRIC MANIFOLDS

  • ERKEN, I. KUPELI
    • Honam Mathematical Journal
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    • v.37 no.4
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    • pp.457-468
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    • 2015
  • The aim of present paper is to investigate 3-dimensional ${\xi}$-projectively flt and $\tilde{\varphi}$-projectively flt normal almost paracontact metric manifolds. As a first step, we proved that if the 3-dimensional normal almost paracontact metric manifold is ${\xi}$-projectively flt then ${\Delta}{\beta}=0$. If additionally ${\beta}$ is constant then the manifold is ${\beta}$-para-Sasakian. Later, we proved that a 3-dimensional normal almost paracontact metric manifold is $\tilde{\varphi}$-projectively flt if and only if it is an Einstein manifold for ${\alpha},{\beta}=const$. Finally, we constructed an example to illustrate the results obtained in previous sections.

INVARIANT MEASURE AND THE EULER CHARACTERISTIC OF PROJECTIVELY ELAT MANIFOLDS

  • Jo, Kyeong-Hee;Kim, Hyuk
    • Journal of the Korean Mathematical Society
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    • v.40 no.1
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    • pp.109-128
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    • 2003
  • In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP$^{n}$ invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that the Chen's conjecture is true for a closed affinely flat manifold whose holonomy group action permits an invariant probability Borel measure on RP$^{n}$ ; that is, such a closed affinly flat manifold has a vanishing Euler characteristic.

On the projectively flat finsler space with a special $(alpha,beta)$-metric

  • Kim, Byung-Doo
    • Communications of the Korean Mathematical Society
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    • v.11 no.2
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    • pp.407-413
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    • 1996
  • The $(\alpha, \beta)$-metric is a Finsler metric which is constructed from a Riemannian metric $\alpha$ and a differential 1-form $\Beta$; it has been sometimes treat in theoretical physics. In particular, the projective flatness of Finsler space with a metric $L^2 = 2\alpha\beta$ is considered in detail.

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PROJECTIVELY FLAT FINSLER SPACES WITH CERTAIN (α, β)-METRICS

  • Park, Hong-Suh;Park, Ha-Yong;Kim, Byung-Doo;Choi, Eun-Seo
    • Bulletin of the Korean Mathematical Society
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    • v.40 no.4
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    • pp.649-661
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    • 2003
  • The ($\alpha,\;\beta$)-metric is a Finsler metric which is constructed from a Riemannian metric $\alpha$ and a differential 1-form $\beta$. In this paper, we discuss the projective flatness of Finsler spaces with certain ($\alpha,\;\beta$)-metrics ([5]) in a locally Minkowski space.

PROJECTIVELY FLAT FINSLER SPACE WITH AN APPROXIMATE MATSUMOTO METRIC

  • Park, Hong-Suh;Lee, Il-Yong;Park, Ha-Yong;Kim, Byung-Doo
    • Communications of the Korean Mathematical Society
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    • v.18 no.3
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    • pp.501-513
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    • 2003
  • The Matsumoto metric is an ($\alpha,\;\bata$)-metric which is an exact formulation of the model of Finsler space. Lately, this metric was expressed as an infinite series form for $$\mid$\beat$\mid$\;<\;$\mid$\alpha$\mid$$ by the first author. He introduced an approximate Matsumoto metric as the ($\alpha,\;\bata$)-metric of finite series form and investigated it in [11]. The purpose of the present paper is devoted to finding the condition for a Finsler space with an approximate Matsumoto metric to be projectively flat.

SEMI-RIEMANNIAN SUBMANIFOLDS OF A SEMI-RIEMANNIAN MANIFOLD WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION

  • Yucesan, Ahmet;Yasar, Erol
    • Communications of the Korean Mathematical Society
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    • v.27 no.4
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    • pp.781-793
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    • 2012
  • We study some properties of a semi-Riemannian submanifold of a semi-Riemannian manifold with a semi-symmetric non-metric connection. Then, we prove that the Ricci tensor of a semi-Riemannian submanifold of a semi-Riemannian space form admitting a semi-symmetric non-metric connection is symmetric but is not parallel. Last, we give the conditions under which a totally umbilical semi-Riemannian submanifold with a semi-symmetric non-metric connection is projectively flat.