where n is the exterior normal direction of ∂Ω. In other words, we look for a “best” way to extend the boundary value φ with the prescribed normal derivative ψ. Typical examples of Ω and N are the unit ball and the unit sphere, respectively. In this case, ψ : ∂Ω → TφN means φ (x) · ψ (x) = 0 for all |x| = 1. With the given Dirichlet data φ, the most natural extension is perhaps the harmonic map...

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