Deformations of Metrics and Biharmonic Maps

Abstract

We construct biharmonic non-harmonic maps between Riemannian manifolds (M, g) and (N, h) by first making the ansatz that φ : (M, g) → (N, h) be a harmonic map and then deforming the metric on N by{\tilde h_\alpha } = \alpha h + \left( {1 - \alpha } \right){\rm{d}}f \otimes {\rm{d}}fto render φ biharmonic, where f is a smooth function with gradient of constant norm on (N, h) and α ∈ (0, 1). We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.

Ricci soliton on a class of Riemannian manifolds under D-isometric deformation.

“…Note that the simplest case for this deformation is for η = df where f ∈ C ∞ (M ). This case has been studied in [4] and [5].…”

Ricci-Soliton on Cosymplectic Manifolds Under Generalized D-Conformal Deformations

“…Ricci-Solitons have been studied by many mathematicians in some different classes of contact geometry since Sharma applied Ricci-Solitons to K-contact manifolds [17]. For more in-depth about recent work on Ricci-Solitons, see [4,9]. A Riemannian manifold (M, g) is called a Ricci soliton if the following condition is satisfied for arbitrary vector fields X, Y on M :…”