• 대한수학회보
• /
• 제37권3호
• /
• pp.543-550
• /
• 2000

We prove that every generalized Landsberg manifold of scalar curvature R is a Riemannian manifold of constant curvature, provided that $R\neq\ 0$.

• 대한수학회보
• /
• 제40권2호
• /
• 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.

• 대한수학회보
• /
• 제47권4호
• /
• pp.767-775
• /
• 2010

Let ${\varphi}\;:\;(M^n,\;F)\;{\rightarrow}\;(\overline{M}^{n+p},\;\overline{F})$ be an isometric immersion from a Finsler manifold to a Finsler manifold. In this paper, we shall obtain the Gauss and Codazzi equations with respect to the Chern connection on submanifolds M, by which we prove that if M is a weakly totally geodesic submanifold of $\overline{M}$, then flag curvature of M equals flag curvature of $\overline{M}$.

• 대한수학회보
• /
• 제40권3호
• /
• pp.457-464
• /
• 2003

Given a Riemannian manifold (M, $\alpha$) with an almost Hermitian structure f and a non-vanishing covariant vector field b, consider the generalized Randers metric $L\;=\;{\alpha}+{\beta}$, where $\beta$ is a special singular Riemannian metric defined by b and f. This metric L is called an (a, b, f)-metric. We compute the inverse and the determinant of the fundamental tensor ($g_{ij}$) of an (a, b, f)-metric. Then we determine the maximal domain D of $TM{\backslash}O$ for an (a, b, f)-manifold where a y-local Finsler structure L is defined. And then we show that any (a, b, f)-manifold is quasi-C-reducible and find a condition under which an (a, b, f)-manifold is C-reducible.

• 대한수학회보
• /
• 제50권3호
• /
• pp.833-837
• /
• 2013

In this note we establish a generalized Myers theorem under line integral curvature bound for Finsler manifolds.

• 대한수학회논문집
• /
• 제14권1호
• /
• 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.

• 호남수학학술지
• /
• 제42권1호
• /
• pp.75-91
• /
• 2020

In this study, we have generalized pseudo-Finsler map by introducing the concept of semi-Riemannian map and have found pseudo-Finsler eikonal equations using pseudo-Finsler map. After that, we have obtained some sufficient theorems on pseudo-Finsler manifolds for the existence of solutions to the eikonal equation. At the same time, we have introduced a natural definition for the affine maps between pseudo-Finsler manifolds and have reached the affine solutions of them.

• 대한수학회보
• /
• 제38권2호
• /
• 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.

• 대한수학회보
• /
• 제42권2호
• /
• 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.

• Korean Journal of Mathematics
• /
• 제14권1호
• /
• pp.79-83
• /
• 2006

In this paper, we apply Zermelo's problem of navigation on Riemannian manifolds to Hermitian manifolds. Using a similar technique with which we define a Randers metric in a Finsler manifold by perturbing Riemannian metric with a vector field, we construct an $(a,b,f)$-metric in a Rizza manifold from a Hermitian metric and a given vector field.