The biharmonicity of sections of the tangent bundle

2014/07/04 by Michael Markellos, Markellos, Michael, Hajime Urakawa +1
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #58E20 #Advanced Differential Geometry Research #Dermatological and Skeletal Disorders #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.1407.1127

openalex publication_date 2014/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The bienergy of a vector field on a Riemannian manifold (M,g) is defined to be the bienergy of the corresponding map (M,g) ---> (TM,g_S), where the tangent bundle TM is equipped with the Sasaki metric g_S. The constrained variational problem is studied, where variations are confined to vector fields, and the corresponding critical point condition characterizes biharmonic vector fields. Furthermore, we prove that if (M,g) is a compact oriented m-dimensional Riemannian manifold and X a tangent vector of M, then X is a biharmonic vector field of (M,g) is and only if X is parallel. Finally, we give examples of non-parallel biharmonic vector fields in the case which the basic manifold (M,g) is non-compact.

Citations

• Harmonic Mappings of Riemannian Manifolds

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