## 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

### Keywords:

group of isometries; Lie groups; Finsler spaces
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