Relation between metric spaces and Finsler spaces |
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Authors: | Lajos Tamássy |
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Institution: | Department of Mathematics, University of Debrecen, 4010 Debrecen, P.O. Box 12, Hungary |
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Abstract: | In a connected Finsler space Fn=(M,F) every ordered pair of points p,q∈M determines a distance ?F(p,q) as the infimum of the arc length of curves joining p to q. (M,?F) is a metric space if Fn is absolutely homogeneous, and it is quasi-metric space (i.e. the symmetry: ?F(p,q)=?F(q,p) fails) if Fn is positively homogeneous only. It is known the Busemann-Mayer relation , for any differentiable curve p(t) in an Fn. This establishes a 1:1 relation between Finsler spaces Fn=(M,F) and (quasi-) metric spaces (M,?F).We show that a distance function ?(p,q) (with the differentiability property of ?F) needs not to be a ?F. This means that the family {(M,?)} is wider than {(M,?F)}. We give a necessary and sufficient condition in two versions for a ? to be a ?F, i.e. for a (quasi-) metric space (M,?) to be equivalent (with respect to the distance) to a Finsler space (M,F). |
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Keywords: | 53C60 51K10 51K05 |
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