• Journal of the Korean Mathematical Society
• /
• v.44 no.1
• /
• pp.237-247
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• v.9 no.3
• /
• pp.649-658
• /
• 1994

The notion of the almost Hermitian Finsler manifold admitting an almost complex structure $f^i_j(x)$ was, for the first time, introduced by G. B. Rizza [8]. It is known that the almost Hermitian Finsler manifold (or a Rizza manifold) has been studied by Y. Ichijyo [2] and H. Hukui [1]. In those papers, the almost Hermitian Finsler manifold which the h-covariant derivative of the almost complex structure $f^i_j(x)$ with respect to the Cartan's Finsler connection vanishes was defined as the Kaehlerian Finsler manifold. The nearly Kaehlerian Finsler manifold was defined and studied by the former of authors [7]. The present paper is the continued study of above paper.

• Bulletin of the Korean Mathematical Society
• /
• v.38 no.2
• /
• pp.303-315
• /
• 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.

• Bulletin of the Korean Mathematical Society
• /
• v.42 no.2
• /
• pp.405-413
• /
• 2005

We consider the connections $\nabla$ on the Rizza manifold (M, J, L) satisfying ${\nabla}G=0\;and\;{\nabla}J=0$. Among them, we derive a Lichnerowicz connection from the Cart an connection and characterize it in terms of torsion. Generalizing Kahler condition in Hermitian geometry, we define a Kahler condition for Rizza manifolds. For such manifolds, we show that the Cartan connection and the Lichnerowicz connection coincide and that the almost complex structure J is integrable.

• Bulletin of the Korean Mathematical Society
• /
• v.49 no.1
• /
• pp.49-55
• /
• 2012

In this paper, we prove that a complex Finsler manifold is a complex locally Minkowski space if and only if it is a strictly K$\ddot{a}$hler-Berwald manifold with zero holomorphic sectional curvature.

• Bulletin of the Korean Mathematical Society
• /
• v.41 no.1
• /
• pp.137-145
• /
• 2004

In [13], we developed a theory of complex Finsler manifolds to investigate the global geometry of complex Finsler manifolds. There we proved a version of Bonnet-Myers' theorem for complex Finsler manifolds with a certain condition on the Finsler metric which is a generalization of the Kahler condition for the Hermitian metric. In this paper, we show that if the holomorphic sectional curvature of M is ${\geq}\;c^2\;>\;0$, then M is simply connected. This is a generalization of the Synge's theorem in the Riemannian geometry and the Tsukamoto's theorem for Kahler manifolds. The main point of the proof lies in how we can circumvent the convex neighborhood theorem in the Riemannian geometry. A second variation formula of arc length for complex Finsler manifolds is also derived.

• Bulletin of the Korean Mathematical Society
• /
• v.40 no.2
• /
• pp.327-340
• /
• 2003

We are devoted to dealing with the conformal theory of a Rizza manifold with a generalized Finsler metric $G_{ij}$ (x,y) and a new conformal invariant non-linear connection $M^{i}$ $_{j}$ (x,y) constructed from the generalized Cern's non-linear connection $N^{i}$ $_{j}$ (x,y) and almost complex structure $f^{i}$ $_{j}$ (x). First, we find a conformal invariant connection ( $M_{j}$ $^{i}$ $_{k}$ , $M^{i}$ $_{j}$ , $C_{j}$ $^{i}$ $_{k}$ ) and conformal invariant tensors. Next, the nearly Kaehlerian (G, M)-structures under conformal change in a Rizza manifold are investigate.

• Communications of the Korean Mathematical Society
• /
• v.14 no.1
• /
• pp.201-210
• /
• 1999

We introduce the notion of the Finsler metrics compatible with f(5,1)-structure and investigate the properties of Finsler space with such metrics.

• Communications of the Korean Mathematical Society
• /
• v.15 no.2
• /
• pp.339-348
• /
• 2000

We introduce the notion of the Finsler metrics compat-ible with a special Riemannian structure f of type (1,1) satisfying f6+f2=0 and investigate the properties of Finsler space with them.