Vertical and complete lifts from a manifold to its cotangent bundle

Abstract

Publisher Summary This chapter considers various methods by which certain types of tensor fields in M can be extended to c T (M) to give useful information about the relationships between the structures of the two manifolds. The chapter focuses on complete lifts of vector fields, tensor fields of type (1, 1) and skewsymmetric tensor fields of type (1, 2). In each of these cases, the complete lift is defined to be a tensor field of the same type as the original. In general, the vertical lift of a tensor field does not have the same type as the original; nevertheless the construction is a useful one. The methods enable to examine the structure of C T( M ) in relation to that of M. In particular, it is shown how almost complex and similar structures on M can be extended to C T(M). Lifts of affine connections in M is also examined, using the idea of a Riemann extension.