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.


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


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