×

Holomorphic tensor fields and linear connections on a second order tangent bundle. (Russian. English summary) Zbl 1216.53019

Summary: 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 \(\Sigma : T^2M \to T^2M\). 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 transferred by a diffeomorphism of the form \(\Sigma\), respectively, into the lift of a tensor field and the lift of a linear connection given on \(M\).

MSC:

53B05 Linear and affine connections
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53A25 Differential line geometry
PDF BibTeX XML Cite