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Let $M$ be a manifold with all structures smooth which admits a metric $g$. Let $\Gamma$ be a linear connection on $M$ such that the associated covariant derivative $\nabla$ satisfies $\nabla g=g\otimes w$ for some 1-form $w$ on $M$. Then one refers to the above setup as a Weyl structure on $M$ and says that the pair $(g,w)$ fits $\Gamma$. If $\sigma: M\to \bbfR$ and if $(g,w)$ fits $\Gamma$, then $(e^{\sigma g}, w+d \sigma)$ fits $\Gamma$. Thus if one thinks of this as a map $g\mapsto w$, then $e^{\sigma g} \mapsto w+d \sigma$.\par In this paper, the author attempts to apply Weyl's idea above to Finsler spaces. A Finsler fundamental function $L:TM\to \bbfR$ satisfies (i) $L(u)>0$ for all $u\in TM$, $u\ne 0$; (ii) $L(\lambda u)= \lambda L(u)$ for all $\lambda\in \bbfR^+$, $u\in T_pM$; (iii) $L$ is smooth except on the zero section; (iv) if $(x,y)$ are the usual coordinates on $TM$, the matrix $g_{ij}= {\partial^2 (1/2L^2) \over\partial y^i \partial y^j}$ is non! (English) |
