On Finsler-Weyl manifolds and connections

Kozma, L.

Proceedings of the 15th Winter School "Geometry and Physics", GDML_Books, (1996), p. [173]-179 / Harvested from

Résumé

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 $\sigma : M\rightarrow \mathbb {R}$ 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 $L:TM\rightarrow \mathbb {R}$ satisfies (i) $L(u)>0$ for all $u\in TM$, $u\ne 0$; (ii) $L(\lambda u)= \lambda L(u)$ for all $\lambda \in \mathbb {R}^+$, $u\in T_pM$; (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!

Détail

MR MR1463519 | Zbl 0905.53016

EUDML-ID : urn:eudml:doc:221155
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