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We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected Berwald space of finite volume and bounded non-positive flag curvature with rank at least two whose universal cover is irreducible is a locally symmetric space or a locally Minkowski space.

In this paper, we introduce a new algebraic notion, weakly symmetric Lie algebras, to give an algebraic description of an interesting class of homogeneous Riemann-Finsler spaces, weakly symmetric Finsler spaces. Using this new definition, we are able to give a classification of weakly symmetric Finsler spaces with dimensions 2 and 3. Finally, we show that all the non-Riemannian reversible weakly symmetric Finsler spaces we find are non-Berwaldian and with vanishing $\text{S}$-curvature. This means that reversible non-Berwaldian Finsler spaces with vanishing $\text{S}$-curvature may exist at large. Hence the generalized volume comparison theorems due to $\text{Z}$. Shen are valid for a rather large class of Finsler spaces.

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Higher rank rigidity for Berwald spaces

An Algebraic Approach to Weakly Symmetric Finsler Spaces

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2 results

Higher rank rigidity for Berwald spaces

An Algebraic Approach to Weakly Symmetric Finsler Spaces