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 $\sigma :M\to ℝ$ 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 $e^{\sigma g}\mapsto w+d\sigma$.In this paper, the author attempts to apply Weyl’s idea above to Finsler spaces. A Finsler fundamental function $L:TM\to ℝ$ satisfies (i) $L\left(u\right)>0$ for all $u\in TM$, $u\ne 0$; (ii) $L\left(\lambda u\right)=\lambda L\left(u\right)$ for all $\lambda \in {ℝ}^{+}$, $u\in T_{p}M$; (iii) $L$ is smooth except on the zero section; (iv) if $(x,y)$ are the usual coordinates on $TM$, the matrix $g_{ij}=\frac{{\partial }^2(1/2L^2)}{\partial y^i\partial y^j}$ is non!

EUDML-ID : urn:eudml:doc:221155
Mots clés:

@article{701584,
     title = {On Finsler-Weyl manifolds and connections},
     booktitle = {Proceedings of the 15th Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {1996},
     pages = {[173]-179},
     mrnumber = {MR1463519},
     zbl = {0905.53016},
     url = {http://dml.mathdoc.fr/item/701584}
}

Kozma, L. On Finsler-Weyl manifolds and connections, dans Proceedings of the 15th Winter School "Geometry and Physics", GDML_Books,  (1996), pp. [173]-179. http://gdmltest.u-ga.fr/item/701584/