Some results on harmonic and bi-harmonic maps. (English) Zbl 1375.53080
Summary: In this paper, we prove that any bi-harmonic map from a compact orientable Riemannian manifold without boundary \((M,g)\) to Riemannian manifold \((N,h)\) is necessarily constant with \((N,h)\) admitting a strongly convex function \(f\) such that \(\mathrm{grad}^Nf\) is a Jacobi-type vector field (or \((N,h)\) admitting a proper homothetic vector field). We also prove that every harmonic map from a complete Riemannian manifold into a Riemannian manifold admitting a proper homothetic vector field, satisfying some condition, is constant. We present an open problem.
MSC:
| 53C43 | Differential geometric aspects of harmonic maps |
| 58E20 | Harmonic maps, etc. |
| 53A30 | Conformal differential geometry (MSC2010) |
Keywords:
harmonic maps; bi-harmonic maps; homothetic vector fields; convex functions; Jacobi-type vector fieldsReferences:
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