Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hopf-type hypersurfaces on Hermite-like manifolds

Mehmet Gulbahar

TL;DR

This work extends Hopf hypersurface theory to Hermite-like statistical manifolds by introducing dual shape operators $A_N$ and $A_N^{\ast}$ and formulating Hopf conditions through Gauss/Weingarten relations. It analyzes concurrent and geodesic vector fields, derives the 1-form $\theta$, and develops curvature relations for tangential hypersurfaces, including explicit expressions for the Riemann curvature tensors $R$ and $R^{\*}$ in the Kaehler-like statistical setting with constant holomorphic curvature $c$. The results reveal how assumptions like $\nabla_X\varphi=0$ constrain the ambient geometry, sometimes forcing $c=0$ or yielding statistical complex space forms, and provide curvature-inequality criteria with sharp equality cases. Collectively, the paper broadens classical Hopf hypersurface geometry to Hermite-like statistics and clarifies how two intertwining connections shape curvature, geodesic behavior, and invariants.

Abstract

The object of this paper is to introduce new classes of hypersurfaces of almost product-like statistical manifolds. The main properties and relations on $K-$para contact, para cosymplectic, para Sasakian and conformal hypersurfaces are obtained. Some examples of these hypersurfaces are presented.

Hopf-type hypersurfaces on Hermite-like manifolds

TL;DR

This work extends Hopf hypersurface theory to Hermite-like statistical manifolds by introducing dual shape operators and and formulating Hopf conditions through Gauss/Weingarten relations. It analyzes concurrent and geodesic vector fields, derives the 1-form , and develops curvature relations for tangential hypersurfaces, including explicit expressions for the Riemann curvature tensors and in the Kaehler-like statistical setting with constant holomorphic curvature . The results reveal how assumptions like constrain the ambient geometry, sometimes forcing or yielding statistical complex space forms, and provide curvature-inequality criteria with sharp equality cases. Collectively, the paper broadens classical Hopf hypersurface geometry to Hermite-like statistics and clarifies how two intertwining connections shape curvature, geodesic behavior, and invariants.

Abstract

The object of this paper is to introduce new classes of hypersurfaces of almost product-like statistical manifolds. The main properties and relations on para contact, para cosymplectic, para Sasakian and conformal hypersurfaces are obtained. Some examples of these hypersurfaces are presented.
Paper Structure (4 sections, 25 theorems, 68 equations)

This paper contains 4 sections, 25 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hopf Hypersurfaces
  4. Riemannian Curvature Tensor Fields on Hopf Hypersurfaces

Key Result

Proposition 3.2

Let $(\widetilde{M},\widetilde{g},\widetilde{\nabla},J)$ be a Kaehler-like statistical manifold and $(M,g)$ be a a Hopf hypersurface of $(\widetilde{M},\widetilde{g},\widetilde{\nabla},J)$. The following relations are satisfied:

Theorems & Definitions (39)

  • Example 2.1
  • Definition
  • Example 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • Definition
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • ...and 29 more