• Journal of the Korean Mathematical Society
• /
• v.48 no.1
• /
• pp.63-82
• /
• 2011

On the hyperbolic space $D^n$, codimension-one totally geodesic foliations of class $C^k$ are classified. Except for the unique parabolic homogeneous foliation, the set of all such foliations is in one-one correspondence (up to isometry) with the set of all functions z : [0, $\pi$] $\rightarrow$ $S^{n-1}$ of class $C^{k-1}$ with z(0) = $e_1$ = z($\pi$) satisfying |z'(r)| ${\leq}1$ for all r, modulo an isometric action by O(n-1) ${\times}\mathbb{R}{\times}\mathbb{Z}_2$. Since 1-dimensional metric foliations on $D^n$ are always either homogeneous or flat (that is, their orthogonal distributions are integrable), this classifies all 1-dimensional metric foliations as well. Equations of leaves for a non-trivial family of metric foliations on $D^2$ (called "fifth-line") are found.

• Honam Mathematical Journal
• /
• v.44 no.3
• /
• pp.432-446
• /
• 2022

This article introduces the structure of GCR-lightlike submanifolds of an indefinite Kaehler statistical manifold and derives their geometric properties. The characterizations on totally geodesic, mixed geodesic, D-geodesic and D'-geodesic GCR-lightlike submanifolds have also been obtained.

• Honam Mathematical Journal
• /
• v.46 no.2
• /
• pp.224-236
• /
• 2024

This study inspects the structure of CR-product of a holomorphic statistical manifold. Findings concerning geodesic submanifolds and totally geodesic foliations in the context of dual connections have been demonstrated. The integrability of distributions in CR-statistical submanifolds has been characterized. The statistical version of CR-product in the holomorphic statistical manifold has been researched. Additionally, some assertions for curvature tensor field of the holomorphic statistical manifold have been substantiated.

• Honam Mathematical Journal
• /
• v.46 no.3
• /
• pp.452-472
• /
• 2024

We introduce the study of a semi-slant lightlike submersion from an indefinite Kaehler manifold onto an r-lightlike manifold. After giving the definition of a semi-slant lightlike submersion, we establish the existence Theorems for this class of lightlike submersions. Then, we derive the integrability conditions for the distributions D₁, D₂ and ∆ associated with a semi-slant lightlike submersion. Consequently, we find some necessary and sufficient conditions for the foliations determined by the distributions D₁, D₂ and ∆. Subsequently, we examine the geometry of totally umbilical fibres of a semi-slant lightlike submersion.