Harmonicity of sections of sphere bundles. (English) Zbl 1161.53050
Summary: We consider the energy functional on the space of sections of a sphere bundle over a Riemannian manifold \((M,\langle \cdot, \cdot \rangle)\) equipped with the Sasaki metric and discuss the characterising condition for critical points. Furthermore, we provide a useful method for computing the tension field in some particular situations. Such a method is shown to be adequate for many tensor fields defined on manifolds \(M\) equipped with a \(G\)-structure compatible with \(\langle \cdot, \cdot \rangle\). This leads to the construction of several new examples of differential forms which are harmonic sections or determine a harmonic map from \((M,\langle \cdot, \cdot \rangle)\) into its sphere bundle.
MSC:
| 53C43 | Differential geometric aspects of harmonic maps |
| 53C10 | \(G\)-structures |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
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