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

TL;DR

This paper studies special vector fields called biharmonic and interpolating sesqui-harmonic on the tangent bundle of a para-Kähler--Norden manifold with the varphi-Sasaki metric, deriving their characterizations and providing illustrative examples.

Contribution

It introduces explicit characterizations of biharmonic and interpolating sesqui-harmonic vector fields in this geometric setting, extending higher-order harmonicity theory.

Findings

Derived first variation formulas for bienergy and interpolating sesqui-energy.

Established explicit conditions for vector fields to be biharmonic or interpolating sesqui-harmonic.

Provided examples illustrating differences between harmonic, biharmonic, and interpolating sesqui-harmonic vector fields.

Abstract

This work investigates biharmonic and interpolating sesqui-harmonic vector fields on the tangent bundle of a para-K\"ahler--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.