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Some results on harmonic maps for Finsler manifolds. (English) Zbl 1085.58012

It is proved that there is no nondegenerate stable harmonic map from the Riemannian unit sphere \(S^n\) (\(n\geq3\)) to any Finsler manifold. Also, any nondegenerate harmonic map from a compact Einstein manifold with nonnegative scalar curvature to a Berwald manifold with nonpositive flag curvature must be totally geodesic. Here a Berwald manifold is a Finsler manifold for which there are natural Christoffel symbols that do not depend on the direction in tangent space.
The proofs exploit the second variation formula and a generalized Weitzenböck formula. The flag curvature enters the statement since it contributes to one term in the second variation.

MSC:

58E20 Harmonic maps, etc.
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
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