• Bulletin of the Korean Mathematical Society
• /
• v.60 no.6
• /
• pp.1607-1620
• /
• 2023

In this paper, we mainly study the problem of the existence of homogeneous geodesics in sub-Finsler manifolds. Firstly, we obtain a characterization of a homogeneous curve to be a geodesic. Then we show that every compact connected homogeneous sub-Finsler manifold and Carnot group admits at least one homogeneous geodesic through each point. Finally, we study a special class of ℓ^p-type bi-invariant metrics on compact semi-simple Lie groups. We show that every homogeneous curve in such a metric space is a geodesic. Moreover, we prove that the Alexandrov curvature of the metric space is neither non-positive nor non-negative.

• Honam Mathematical Journal
• /
• v.30 no.2
• /
• pp.335-341
• /
• 2008

In this paper, we use the convexity of distance function between geodesics in a singular Hadamard space to generalize Hadamard-Cartan theorem for 2-dimensional metric spaces. We also determine a neighborhood of a closed geodesic where no other closed geodesic exists in a complete space of nonpositive curvature.

• Communications of the Korean Mathematical Society
• /
• v.10 no.1
• /
• pp.163-173
• /
• 1995

Inspired in [2,9,10,17], pp. E. Ehrlich and S. B. Kim in [4] cultivated the Riccati equation related to the Raychaudhuri equation of General Relativity for the stable Jacobi tensor along the geodesics to extend the Hawking-Penrose conjugacy theorem to $$ f(t) = Ric(c(t)',c'(t)) + tr(\sigma(A)^2) $$ where $\sigma(A)$ is the shear tensor of A for the stable Jacobi tensor A with $A(t_0) = Id$ along the complete Riemannian or complete nonspacelike geodesics c.

• Journal of the Korean Mathematical Society
• /
• v.40 no.6
• /
• pp.951-962
• /
• 2003

In this paper we address focal points and treat manifolds (M, g) whose Lorentzian metric tensors g have a spacelike $C^{0}$-hypersurface $\Sigma$ [10]. We apply Jacobi fields for such manifolds, and check the local length maximizing properties of $C^1$-geodesics. The condition of maximality of timelike curves(geodesics) passing $C^{0}$-hypersurface is studied.ied.

• Bulletin of the Korean Mathematical Society
• /
• v.37 no.2
• /
• pp.337-346
• /
• 2000

The geodesic equation in a two-dimensional Finsler space is given by the differential equation of the Weierstrass form. In the present paper, we express the differential equations of geodesics in a two-dimensional Finsler space with a generalized Kropina metric.

• Journal of the Korean Mathematical Society
• /
• v.57 no.1
• /
• pp.195-213
• /
• 2020

We define two types of null hypersurfaces as; isoparametric and quasi isoparametric null hypersurfaces of Lorentzian space forms, based on the two shape operators associated with a null hypersurface. We prove that; on any screen conformal isoparametric null hypersurface, the screen geodesics lie on circles in the ambient space. Furthermore, we prove that the screen distributions of isoparametric (or quasi isoparametric) null hypersurfaces with at most two principal curvatures are generally Riemannian products. Several examples are also given to illustrate the main concepts.

• East Asian mathematical journal
• /
• v.26 no.5
• /
• pp.607-614
• /
• 2010

In this paper, we obtain geodesic formula of a certain class of Pseudoriemmanian 2-step nilpotent groups and show a constancy of represenation matrix of Jacobi oprerators along geodesics in Pseudoriemmanian 2-step nilpotent groups with one dimensional center.

• Kyungpook Mathematical Journal
• /
• v.63 no.1
• /
• pp.79-95
• /
• 2023

In this note, we define a Berger type deformed Sasaki metric as a natural metric on the second tangent bundle of a manifold by means of a biparacomplex structure. First, we obtain the Levi-Civita connection of this metric. Secondly, we get the curvature tensor, sectional curvature, and scalar curvature. Afterwards, we obtain some formulas characterizing the geodesics with respect to the metric on the second tangent bundle. Finally, we present the harmonicity conditions for some maps.

• Communications of the Korean Mathematical Society
• /
• v.6 no.1
• /
• pp.73-87
• /
• 1991

• Communications of the Korean Mathematical Society
• /
• v.3 no.2
• /
• pp.249-256
• /
• 1988