On Finsler-Weyl manifolds and connections

Kozma, L.. "On Finsler-Weyl manifolds and connections." Proceedings of the 15th Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 1996. [173]-179. <http://eudml.org/doc/221155>.

@inProceedings{Kozma1996,
abstract = {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!},
author = {Kozma, L.},
booktitle = {Proceedings of the 15th Winter School "Geometry and Physics"},
keywords = {Proceedings; Geometry; Physics; Winter school; Srni (Czech Republic)},
location = {Palermo},
pages = {[173]-179},
publisher = {Circolo Matematico di Palermo},
title = {On Finsler-Weyl manifolds and connections},
url = {http://eudml.org/doc/221155},
year = {1996},
}

TY - CLSWK
AU - Kozma, L.
TI - On Finsler-Weyl manifolds and connections
T2 - Proceedings of the 15th Winter School "Geometry and Physics"
PY - 1996
CY - Palermo
PB - Circolo Matematico di Palermo
SP - [173]
EP - 179
AB - 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!
KW - Proceedings; Geometry; Physics; Winter school; Srni (Czech Republic)
UR - http://eudml.org/doc/221155
ER -