Geodesic
Holomorphic Tensor Fields and Linear Connections on a Second Order Tangent Bundle
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 151 (2009) no. 4, pp. 36-50
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The second order tangent bundle $T^2M$ of a smooth manifold $M$ carries a natural structure of a smooth manifold over the algebra $\mathbf R(\varepsilon^2)$ of truncated polynomials of degree 2. A section $\sigma$ of $T^2M$ induces an $\mathbf R(\varepsilon^2)$-smooth diffeomorphism $\Sigma\colon T^2M\to T^2M$. Conditions are obtained under which an $\mathbf R(\varepsilon^2)$-smooth tensor field and an $\mathbf R(\varepsilon^2)$-smooth linear connection on $T^2M$ can be transfered by a diffeomorphism of the form $\Sigma$, respectively, into the lift of a tensor field and the lift of a linear connection given on $M$.
Keywords: tangent bundle of second order, lift of a linear connection, lift of a tensor field, holomorphic connection, Lie derivative. 
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     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
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F. R. Gainullin; V. V. Shurygin. Holomorphic Tensor Fields and Linear Connections on a Second Order Tangent Bundle. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 151 (2009) no. 4, pp. 36-50. http://geodesic.mathdoc.fr/item/UZKU_2009_151_4_a3/

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