View source for Weyl-Otsuki space

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''Otsuki–Weyl space''

An Otsuki space [[#References|[a6]]], [[#References|[a7]]] is a [[Manifold|manifold]] $M$ endowed with two different linear connections $\square ^ { '' } \Gamma$ and $\square ^ { \prime } \Gamma$ (cf. also [[Connections on a manifold|Connections on a manifold]]) and a non-degenerate $( 1,1 )$ tensor field $P$ of constant rank (cf. also [[Tensor analysis|tensor analysis]]), where the connection coefficients $\square ^ { \prime \prime } \Gamma _ { j k } ^ { i } ( x )$, $x \in M$, are used in the computation of the contravariant, and the $\square ^ { \prime } \Gamma _ { j k } ^ { i } ( x )$ in the computation of the covariant, components of the invariant (covariant) differential of a tensor (vector). For a tensor field $T$ of type $( 1,1 )$, the invariant differential $DT$ and the covariant differential $\nabla T$ have the following forms

\begin{equation*} D T _ { j } ^ { i } = \nabla _ { k } T _ { j } ^ { i } d x ^ { k } = \end{equation*}

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$\square ^ { \prime } \Gamma$ and $\square ^ { '' } \Gamma$ are connected by the relation

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Thus, $\square ^ { '' } \Gamma$ and $P$ determine $\square ^ { \prime } \Gamma$. T. Otsuki calls these a general connection. For $P ^ { i } _ { r } = \delta ^ { i }_r$ one obtains $\square ^ { \prime } \Gamma = \square ^ { \prime \prime } \Gamma$ and the usual invariant differential.

If $M$ is endowed also with a [[Riemannian metric|Riemannian metric]] $g$, then $\square ^ { \prime \prime } \Gamma _ { j k } ^ { i } ( x )$ may be the [[Christoffel symbol|Christoffel symbol]] $\{ \square _ { j k } ^ { i } \}$.

In a Weyl space $W ^ { n } = ( M , g , \gamma )$ one has $\nabla _ { i g j k } = \gamma _ { i  g  j k }$. A Weyl–Otsuki space $W - O _ { n }$ [[#References|[a1]]] is a $W ^ { m }$ endowed with an Otsuki connection. The $\square ^ { \prime \prime } \Gamma _ { r k } ^ { t }$ are defined here as

\begin{equation*} \square ^ { \prime \prime } \Gamma _ { r k } ^ { t } = \{ \square _ { r k } ^ { t } \} - \frac { 1 } { 2 } g ^ { t s } ( \gamma _ { k } m _ { r s } + \gamma _ { r } m _ { s k } - \gamma _ { s } m _ { r k } ), \end{equation*}

\begin{equation*} m _ { r s } = g _ { ij} Q _ { r } ^ { i } Q _ { s } ^ { j }, \end{equation*}

where $Q$ is the inverse of $P$. $W - O _ { n }$ spaces were studied mainly by A. Moór [[#References|[a2]]], [[#References|[a3]]].

He extended the Otsuki connection also to affine and metrical line-element spaces, obtaining Finsler–Otsuki spaces $F - O _ { n }$ [[#References|[a4]]], [[#References|[a5]]] with invariant differential

\begin{equation*} D T_j^i = \end{equation*}

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Here, all objects depend on the line-element $( x , \dot { x } )$, the $T$, $P$, $\square ^ { \prime } \Gamma$, $\square ^ { '' } \Gamma$ are homogeneous of order $O$, and $C$ is a tensor.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  A. Moór,   "Otsukische Übertragung mit rekurrenter Maß tensor"  ''Acta Sci. Math.'' , '''40'''  (1978)  pp. 129–142</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  A. Moór,   "Über verschiedene geodätische Abweichungen in Weyl–Otsukischen Räumen"  ''Publ. Math. Debrecen'' , '''28'''  (1981)  pp. 247–258</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  A. Moór,   "Über Transformationsgruppen in Weyl–Otsukischen Räumen"  ''Publ. Math. Debrecen'' , '''29'''  (1982)  pp. 241–250</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  A. Moór,   "Über die Begründung von Finsler–Otschukischen Räumen und ihre Dualität"  ''Tensor N.S.'' , '''37'''  (1982)  pp. 121–129</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A. Moór,   "Über spezielle Finsler–Otsukische Räume"  ''Publ. Math. Debrecen'' , '''31'''  (1984)  pp. 185–196</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  T. Otsuki,   "On general connections. I"  ''Math. J. Okayama Univ.'' , '''9'''  (1959-60)  pp. 99–164</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  T. Otsuki,   "On metric general connections"  ''Proc. Japan Acad.'' , '''37'''  (1961)  pp. 183–188</td></tr></table>