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The group of isometries of a Finsler space. (English) Zbl 1055.53055

The main theorem is: Let \((M,F)\) be Finsler space and \(\phi \) a distance preserving mapping of \(M\) onto itself. Then \(\phi \) is a diffeomorphism. A mapping \(\phi \) of \(M\) onto itself is called an isometry if \(\phi \) is a diffeomorphism and for any \(x\in M\), \(X\in T_{x}(M)\), \(F(\phi (x),d\phi_{x}(X)) = F(x,X)\). Among others it is proved that the group of isometries \(I(M)\) of \(M\) is a Lie transformation group of \(M\). If \(I_{x}(M)\) is the isotropy subgroup of \(I(M)\) at \(x\), then \(I_{x}(M)\) is compact.

MSC:

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
54F65 Topological characterizations of particular spaces
54E40 Special maps on metric spaces
54E35 Metric spaces, metrizability
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