• Journal of the Korean Mathematical Society
• /
• v.44 no.6
• /
• pp.1291-1312
• /
• 2007

We study some extremal problems on the Cartan-Hartogs domains. Through computing the minimal circumscribed Hermitian ellipsoid of the Cartan-Hartogs domains, we get the $Carath\acute{e}odory$ extremal mappings between the Cartan-Hartogs domains and the unit hyperball, and the explicit formulas for computing the $Carath\acute{e}odory$ extremal value.

• Honam Mathematical Journal
• /
• v.45 no.4
• /
• pp.610-618
• /
• 2023

In this paper, we obtain (k, m)-type slant helices for a null Cartan curve with the Bishop frame in Minkowski space E⁴₁.

• Journal of the Korean Mathematical Society
• /
• v.38 no.2
• /
• pp.385-408
• /
• 2001

This is an introduction to a stochastic version of E. Cartan′s symplectic mechanics. A class of time-symmetric("Bernstein") diffusion processes is used to deform stochastically the exterior derivative of the Poincare-Cartan one-form on the extended phase space. The resulting symplectic tow-form is shown to contain the (a.e.) dynamical laws of the diffusions. This can be regarded as a geometrization of Feynman′s path integral approach to quantum theory; when Planck′s constant reduce to zero, we recover Cartan′s mechanics. The underlying strategy is the one of "Euclidean Quantum Mechanics".

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.3
• /
• pp.885-899
• /
• 2013

In this paper, we study null helices, null slant helices and Cartan slant helices in $\mathb{E}^3_1$. Using some associated curves, we characterize the null helices and the Cartan slant helices and construct them. Also, we study a space-like curve with the principal normal vector field which is a degenerate plane curve.

• East Asian mathematical journal
• /
• v.35 no.5
• /
• pp.529-534
• /
• 2019

The purpose of this article is to present The $Carath{\acute{e}}odory$-Cartan-Kaup-Wu theorem for connected Kobayashi hyperbolic almost complex manifolds.

• Honam Mathematical Journal
• /
• v.40 no.3
• /
• pp.561-576
• /
• 2018

In this paper, we define null Cartan and pseudo null Bertrand curves in Minkowski space ${\mathbb{E}}^3_1$ according to their Bishop frames. We obtain the necessary and sufficient conditions for pseudo null curves to be Bertand B-curves in terms of their Bishop curvatures. We prove that there are no null Cartan curves in Minkowski 3-space which are Bertrand B-curves, by considering the cases when their Bertrand B-mate curves are spacelike, timelike, null Cartan and pseudo null curves. Finally, we give some examples of pseudo null Bertrand B-curve pairs.

• Journal of the Korean Mathematical Society
• /
• v.34 no.4
• /
• pp.771-790
• /
• 1997

By using the method of equivalence of E. Cartan we calculate the local scalar invariants for Riemannian 2-maniolds. We define also a notion of local invariants for submanifolds in $R^{n + d}, n \geq 2, d \geq 1$, in terms of the symmetry of the local isometric embedding equations of Riemannian n-manifolds into $R^{n + d}$. We show that the local invariants obtained by the Cartan's method are the intrinsic expressions of the local invariants in our sense in the casees of surfaces in $R^3$.

• Journal for History of Mathematics
• /
• v.22 no.2
• /
• pp.13-26
• /
• 2009

Elie Cartan is one of the mathematicians whose works marked the development of geometry and algebra of 20th century. We illuminate his mathematical contributions which is familiar to the experts but is alien to most of the mathematicians in general. Also we elucidate some of his influences.

• Journal of the Korean Mathematical Society
• /
• v.54 no.6
• /
• pp.1683-1698
• /
• 2017

In this paper, we study the $Carath{\acute{e}}odory$ extremal problems on the Hua domain of the first three types. We give the explicit formula for the $Carath{\acute{e}}odory$ extremal problems between the first three types of Hua domain and the unit ball, which improves the works done on Hua domain and Cartan-egg domain and super-Cartan domain.

• Honam Mathematical Journal
• /
• v.40 no.1
• /
• pp.47-59
• /
• 2018

We define the $e^{\alpha}_1e^{\alpha}_3$-isotropic Smarandache curves of type-1 and type-2, the $e^{\alpha}_1e^{\alpha}_2e^{\alpha}_3$-isotropic Smarandache curve, and the $e^{\alpha}_1e^{\alpha}_2e^{\alpha}_4$-isotropic Smarandache curves of type-1 and type-2. Then we examine these kinds of isotropic Smarandache curve according to Cartan frame in the complex 4-space $\mathbb{C}^4$ and give some differential geometric properties of these Samarandache curves.