• Title, Summary, Keyword: holomorphic sectional curvature

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KÄHLER SUBMANIFOLDS WITH LOWER BOUNDED TOTALLY REAL BISECTIONL CURVATURE TENSOR II

  • Pyo, Yong-Soo;Shin, Kyoung-Hwa
    • Communications of the Korean Mathematical Society
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    • v.17 no.2
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    • pp.279-293
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    • 2002
  • In this paper, we prove that if every totally real bisectional curvature of an n($\geq$3)-dimensional complete Kahler submanifold of a complex projective space of constant holomorphic sectional curvature c is greater than (equation omitted) (3n$^2$+2n-2), then it is totally geodesic and compact.

CONNECTIONS ON ALMOST COMPLEX FINSLER MANIFOLDS AND KOBAYASHI HYPERBOLICITY

  • Won, Dae-Yeon;Lee, Nany
    • Journal of the Korean Mathematical Society
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    • v.44 no.1
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    • pp.237-247
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    • 2007
  • In this paper, we establish a necessary condition in terms of curvature for the Kobayashi hyperbolicity of a class of almost complex Finsler manifolds. For an almost complex Finsler manifold with the condition (R), so-called Rizza manifold, we show that there exists a unique connection compatible with the metric and the almost complex structure which has the horizontal torsion in a special form. With this connection, we define a holomorphic sectional curvature. Then we show that this holomorphic sectional curvature of an almost complex submanifold is not greater than that of the ambient manifold. This fact, in turn, implies that a Rizza manifold is hyperbolic if its holomorphic sectional curvature is bounded above by -1.

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.

LOWER HOUNDS ON THE HOLOMORPHIC SECTIONAL CURVATURE OF THE BERGMAN METRIC ON LOCALLY CONVEX DOMAINS IN $C^{n}$

  • Cho, Sang-Hyun;Lim, Jong-Chun
    • Bulletin of the Korean Mathematical Society
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    • v.37 no.1
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    • pp.127-134
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    • 2000
  • Let $\Omega$ be a bounded pseudoconvex domain in$C^{n}$ with smooth defining function r and let$z_0\; {\in}\; b{\Omega}$ be a point of finite type. We also assume that $\Omega$ is convex in a neighborhood of $z_0$. Then we prove that all the holomorphic sectional curvatures of the Bergman metric of $\Omega$ are bounded below by a negative constant near $z_0$.

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ON THE BONNET′S THEOREM FOR COMPLEX FINSLER MANIFOLDS

  • Won, Dae-Yeon
    • Bulletin of the Korean Mathematical Society
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    • v.38 no.2
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    • pp.303-315
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    • 2001
  • In this paper, we investigate the topology of complex Finsler manifolds. For a complex Finsler manifold (M, F), we introduce a certain condition on the Finsler metric F on M. This is a generalization of Kahler condition for the Hermitian metric. Under this condition, we can produce a Kahler metric on M. This enables us to use the usual techniques in the Kahler and Riemannian geometry. We show that if the holomorphic sectional curvature of $ M is\geqC^2>0\; for\; some\; c>o,\; then\; diam(M)\leq\frac{\pi}{c}$ and hence M is compact. This is a generalization of the Bonnet\`s theorem in the Riemannian geometry.

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