Connections in the second order tangent bundle with extended structure group

Abstract

The second order tangent bundle T ² M of a smooth manifold M carries a natural structure of a smooth manifold over the algebra D ² of truncated polynomials of degree two in one variable, which gives rise to an extended structure group of T ² M and the corresponding extended second order frame bundle $\hat P^2 M$ associated to T ² M. Two connections in $\hat P^2 M$ are said to be equivalent if one of them can be mapped into the other by a fiber preserving D ²-diffeomorphism of T ² M to itself. We establish necessary and sufficient conditions under which two connections in $\hat P^2 M$ are equivalent and in particular the conditions under which a connection in $\hat P^2 M$ is equivalent to a second order differential connection onM.