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**Biharmonic maps from Finsler spaces.**
*(English)*
Zbl 1340.53042

The study of the different kinds of topic-harmonic maps on Riemannian manifolds has a long history. Biharmonic maps on Riemannian manifolds were studied by several authors; see [S. Montaldo and C. Oniciuc, Rev. Unión Mat. Argent. 47, No. 2, 1–22 (2006; Zbl 1140.58004)]. The study of harmonic maps on Finsler manifolds was initiated by X. Mo [Ill. J. Math. 45, No. 4, 1331–1345 (2001; Zbl 0996.53047)] and further discussed by X. Mo and Y. Yang [Sci. China, Ser. A 48, No. 1, 115–130 (2005; Zbl 1127.58011)]. Moreover, an extensive study of harmonic maps of Finsler manifolds was carried on by S. Nishikawa [Topics in differential geometry. Bucharest: Editura Academiei Române, 308 p., 207–247 (2008; Zbl 1157.53032)]. In the paper under review, the author begins with the notion of bi-energy and its first variation, and studies biharmonic maps from a Finsler space to a Riemannian space. The author first extends the concept of the bi-energy functional from Finsler to Riemannian mappings and then determines its Euler-Lagrange equations, i.e., the equations from Finsler to Riemannian biharmonic maps. It is shown that any harmonic map from a Finsler to a Riemannian space is biharmonic but the converse is not true in general. A biharmonic map which is not harmonic, is said to be proper biharmonic. It is, however, worthy to note that two notable results in Riemannian geometry generalized to Finsler manifold (given as under) are proved by the author:

1) If \((M,g)\) is a compact Finsler manifold without boundary and \((\tilde{M},\tilde{g})\) is a Riemannian manifold with non-positive sectional curvature then any biharmonic map \(\phi:M\rightarrow\tilde{M}\) is harmonic.

2) Let \((M,g)\) be an arbitrary Finsler manifold (not necessarily compact), \((\tilde{M},\tilde{g})\) a Riemannian manifold with strictly negative sectional curvature and \(\phi:M\rightarrow\tilde{M}\) be a biharmonic map. If \(\phi\) has the properties:

a) \(| | \tau(\phi)| | =\mathrm{const}\), and

b) there exists a point \(x_{0}\in M\) at which the rank of \(\phi\) is at least 2,

then \(\phi\) is harmonic. Subsequently, the author studies biharmonicity of the identity map \(\mathrm{id}:(M,g)\rightarrow(M,\tilde{g})\) and exhibits that if some non-zero vector field \(A=a^{i}(x){\partial_i}\) is parallel with respect to \(\tilde{g}\), then the identity map is proper biharmonic. In the end, the author discusses the second variation of the bi-energy.

1) If \((M,g)\) is a compact Finsler manifold without boundary and \((\tilde{M},\tilde{g})\) is a Riemannian manifold with non-positive sectional curvature then any biharmonic map \(\phi:M\rightarrow\tilde{M}\) is harmonic.

2) Let \((M,g)\) be an arbitrary Finsler manifold (not necessarily compact), \((\tilde{M},\tilde{g})\) a Riemannian manifold with strictly negative sectional curvature and \(\phi:M\rightarrow\tilde{M}\) be a biharmonic map. If \(\phi\) has the properties:

a) \(| | \tau(\phi)| | =\mathrm{const}\), and

b) there exists a point \(x_{0}\in M\) at which the rank of \(\phi\) is at least 2,

then \(\phi\) is harmonic. Subsequently, the author studies biharmonicity of the identity map \(\mathrm{id}:(M,g)\rightarrow(M,\tilde{g})\) and exhibits that if some non-zero vector field \(A=a^{i}(x){\partial_i}\) is parallel with respect to \(\tilde{g}\), then the identity map is proper biharmonic. In the end, the author discusses the second variation of the bi-energy.

Reviewer: Om Prakash Singh (Agra)