Proper biharmonic maps on tangent bundle

TL;DR

This paper introduces the Mus-Gradient metric on tangent bundles, explores its geometric properties, and constructs examples of proper biharmonic maps, expanding understanding of biharmonic map theory in Riemannian geometry.

Contribution

It defines a new metric on tangent bundles and characterizes a novel class of proper biharmonic maps, with explicit Euclidean examples.

Findings

Characterization of proper biharmonic maps under the Mus-Gradient metric

Construction of explicit Euclidean examples

Geometric analysis of the Mus-Gradient metric

Abstract

This paper, we define the Mus-Gradient metric on tangent bundle $TM$ by a deformation non-conform of Sasaki metric over an n-dimensional Riemannian manifold $(M, g)$. First we investigate the geometry of the Mus-Gradient metric and we characterize a new class of proper biharmonic maps. Examples of proper biharmonic maps are constructed when all of the factors are Euclidean spaces.