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Biharmonic and Interpolating Sesqui-Harmonic Vector Fields with Respect to the varphi-Sasakian Metric

Abderrahim Zagane, Kheireddine Biroud, Medjahed Djilali

TL;DR

The paper studies biharmonic and interpolating sesqui-harmonic vector fields on the tangent bundle of a para-Kähler–Norden manifold $(M^{2m},\varphi,g)$ endowed with the $\varphi$-Sasaki metric $g^{\varphi}$. By deriving the first variations of energy, bienergy, and the interpolating sesqui-energy restricted to vector-field maps, it provides explicit necessary and sufficient conditions for a vector field to be biharmonic or interpolating sesqui-harmonic, including detailed expressions for the tension and bitension fields in this geometry. It also presents concrete examples illustrating when harmonic, biharmonic, and interpolating sesqui-harmonic behaviors coincide or differ, and discusses both compact and non-compact settings. Overall, the work extends higher-order harmonicity theory in pseudo-Riemannian geometry by clarifying how the base’s para-Kähler–Norden structure and the $\varphi$-Sasaki metric influence vector-field variational properties on the tangent bundle.

Abstract

This work investigates biharmonic and interpolating sesqui-harmonic vector fields on the tangent bundle of a para-Kähler--Norden manifold (M, varphi, g) endowed with the varphi-Sasaki metric. We derive the first variation of the bienergy and interpolating sesqui-energy functionals, restricted to the space of vector fields. Explicit characterizations are established for vector fields satisfying the corresponding variational conditions-namely, biharmonicity and interpolating sesqui-harmonicity. Furthermore, several examples are presented to illustrate the general theory and to elucidate the distinctions between harmonic, biharmonic, and interpolating sesqui-harmonic behaviors. These results extend and complement existing research on higher-order harmonicity in pseudo-Riemannian geometry.

Biharmonic and Interpolating Sesqui-Harmonic Vector Fields with Respect to the varphi-Sasakian Metric

TL;DR

The paper studies biharmonic and interpolating sesqui-harmonic vector fields on the tangent bundle of a para-Kähler–Norden manifold endowed with the -Sasaki metric . By deriving the first variations of energy, bienergy, and the interpolating sesqui-energy restricted to vector-field maps, it provides explicit necessary and sufficient conditions for a vector field to be biharmonic or interpolating sesqui-harmonic, including detailed expressions for the tension and bitension fields in this geometry. It also presents concrete examples illustrating when harmonic, biharmonic, and interpolating sesqui-harmonic behaviors coincide or differ, and discusses both compact and non-compact settings. Overall, the work extends higher-order harmonicity theory in pseudo-Riemannian geometry by clarifying how the base’s para-Kähler–Norden structure and the -Sasaki metric influence vector-field variational properties on the tangent bundle.

Abstract

This work investigates biharmonic and interpolating sesqui-harmonic vector fields on the tangent bundle of a para-Kähler--Norden manifold (M, varphi, g) endowed with the varphi-Sasaki metric. We derive the first variation of the bienergy and interpolating sesqui-energy functionals, restricted to the space of vector fields. Explicit characterizations are established for vector fields satisfying the corresponding variational conditions-namely, biharmonicity and interpolating sesqui-harmonicity. Furthermore, several examples are presented to illustrate the general theory and to elucidate the distinctions between harmonic, biharmonic, and interpolating sesqui-harmonic behaviors. These results extend and complement existing research on higher-order harmonicity in pseudo-Riemannian geometry.
Paper Structure (5 sections, 21 theorems, 111 equations)

This paper contains 5 sections, 21 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Basic properties of tangent bundle with the $\varphi$-Sasaki metric
  3. Basic properties of Energy and bienergy functional
  4. Biharmonicity of vector field $\xi:(M^{2m},\varphi,g)\to (TM,g^{\varphi})$
  5. Interpolating sesqui-harmonicity of vector field $\xi:(M^{2m},\varphi,g)\to (TM,g^{\varphi})$

Key Result

Theorem 2.2

Given a para-Kähler--Norden manifold $(M^{2m}, \varphi, g)$ and its tangent bundle $(TM,g^{\varphi})$ endowed with the $\varphi$-Sasaki metric. Then, the Levi-Civita connection $\widetilde{\nabla}$ satisfies: for all vector fields $W$ and $Z$ on $M$, where $\nabla$ is the Levi-Civita connection of $(M^{2m}, \varphi, g)$ and $\mathrm{R}$ is its curvature tensor.

Theorems & Definitions (40)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • Corollary 4.3
  • Corollary 4.4
  • Definition 4.5
  • Definition 4.6
  • ...and 30 more