The prolongations of a Finsler metric to the tangent bunde $T^k(M) (k>1)$ of the higher order accelerations 2005jbq | Atanasiu Gh. // Department of Algebra and Geometry, Transilvania
University, Brasov, Romania
An old problem in differential geometry is that of prolongation of a
Riemannian structure $g\left( x\right) $ on a real $n-$dimensional $%
C^{\infty }$-manifold $M,x\in M,$ to the bundle of $k-$jets $\left(
J_{0}^{k}M,\pi ^{k},M\right) $ or, equivalently the tangent bundle $\left(
T^{k}M,\pi ^{k},M\right) $ of the higher order accelerations. The problem
belongs to so-called geometry of higher order. It was solved in $\left[ 18%
\right] $ for $k=1$ and partially in $\left[ 19\right] $ for$\;k=2.$ The
same problem of prolongation can be considered for a Finslerian structure $%
F\left( x,y^{\left( 1\right) }\right) $. In the paper $\left[ 15\right] $
are given these solutions in the general cases, using the Sasaki-Matsumoto $N-$lift (for $k=2,$ see $\left[ 3\right] $ and $\left[ 6\right] ).
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