Abstract
In this chapter, we study the symmetry of Finsler spaces. Section 5.1 is devoted to studying the geometric properties of locally affine symmetric and globally affine symmetric Berwald spaces. In Sect. 5.2, we prove that any globally symmetric Finsler space must be a Berwald space. In Sect. 5.3, we introduce the notion of a Minkowski symmetric Lie algebra to give an algebraic description of affine symmetric Berwald spaces as well as globally symmetric Finsler spaces. As an important application, we generalize a classical result of E. Cartan from Riemannian geometry to the Finslerian case. In Sect. 5.4, we prove some interesting rigidity results on symmetric Finsler spaces. Finally, in Sect. 5.1, we study complex structures on symmetric Finsler spaces and obtain a complete classification of the complex symmetric Finsler spaces.
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