Deformation quantization of almost Kähler models and Lagrange-Finsler spaces

For coordinates $uα=(xi,ya)$ on $TM$⁠, when indices $i,j,…,a,…,b,…$ run values $1,2,…,n$⁠, we get also coordinates on $TM˜$ if not all fiber coordinates $ya$ vanish; in brief, we shall write $u=(x,y)$⁠.

Proofs consist of straightforward computations.

For simplicity, in this work, we shall omit left labels $L$ in formulas if it will not result in ambiguities; we shall use boldface indices for spaces and objects provided or adapted to an $N$-connection structure.

We contract horizontal and vertical indices following the rule $i=1$ is $a=n+1$⁠; $i=2$ is $a=n+2;…$⁠; $i=n$ is $a=n+n″$⁠.

We chose a definition of this tensor as in Ref. 6 which is with minus sign compared to the definition used in Ref. 21; we also use a different letter for this tensor, like for the $N$-connection curvature, because in our case, the symbol $N$ is used for $N$ connections.

This formula is a nonholonomic analog, for our conventions, with inverse sign, of formula (2.9) from Ref. 21.

It should be noted that formulas (20)–(24) can be written for any metric $g$ and metric compatible $d$ connection $D$⁠, $Dg=0$⁠, on $TM$⁠, provided with arbitrary $N$ connection $N$ (we have to omit hats and labels $L$⁠). It is a more sophisticated problem to define such constructions for Finsler geometries with the so-called Chern connection which are metric noncompatible (Ref. 51). For applications in standard models of physics, we chose the variants of Lagrange-Finsler spaces defined by metric compatible $d$ connections, see discussion in Ref. 13.