He, Qun; Shen, Yibing Some results on harmonic maps for Finsler manifolds. (English) Zbl 1085.58012 Int. J. Math. 16, No. 9, 1017-1031 (2005). 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. Reviewer: Andreas Gastel (Düsseldorf) Cited in 2 ReviewsCited in 17 Documents MSC: 58E20 Harmonic maps, etc. 53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics) Keywords:Finsler manifold; harmonic map; stability; second variation formula PDF BibTeX XML Cite \textit{Q. He} and \textit{Y. Shen}, Int. J. Math. 16, No. 9, 1017--1031 (2005; Zbl 1085.58012) Full Text: DOI References: [1] DOI: 10.1007/978-1-4612-1268-3 [2] DOI: 10.1090/conm/196/02429 [3] DOI: 10.1090/cbms/050 [4] Mo X. H., Sci. China A 41 pp 900– [5] Mo X. H., Illinois J. Math. 45 pp 1331– [6] Pan Y. L., Chinese Ann. Math. 3 pp 515– [7] DOI: 10.1142/9789812811622 [8] Shen Y. B., Sci. China A 47 pp 39– [9] DOI: 10.1215/S0012-7094-80-04736-5 · Zbl 0513.58019 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.