Holomorphic Tensor Fields and Linear Connections on a Second Order Tangent Bundle

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 $M$.