Liftings of Some Types of Tensor Fields and Connections to Tangent Bundles of p^r-Velocities

Abstract

In the previous paper [6], we studied the liftings of tensor fields to tangent bundles of higher order. The purpose of the present paper is to , λ p ) of non-negative integers λi satisfying ΣΛ ^ r. In § 2, we construct <£>-lifts of any vector fields and U)-lifts of 1-forms. The <J>-lift is a little bit different from the U)-lift of vector fields in [6].In §3, we construct (X)-lifting of (0, #)-tensor fields and then (Λ)-lifting r, V of (1, #)-tensor fields to TM for q^l. Unfortunately, the author could not r, v construct a natural lifting of (s, #)-tensor fields to TM for s;>:2. Nevertheless, our (^)-liftings of (s, g)-tensor fields for 5 = 0 or 1 are quite sufficient for the geometric applications, because the important tensor fields with which we encounter so far in differential geometry seem to be, fortunately, only of type (s, q) with s = 0 or 1.As an application, we shall consider in § 4, the prolongations of almost complex structures and prove that if M is a (homogeneous) complex manifold, r, p then TM is also a (homogeneous) complex manifold. r,pIn § 5, we consider the liftings of affine connections to TM and prove r, p that if M is locally affine symmetric then TM is also locally affine symmetric with respect to the lifted affine connection.In § 6, we shall give a proof for the fact that if M is an affine symr, p metric space then TM is also an affine symmetric space.

NATURAL LIFTINGS OF CONNECTIONS TO THE r-th ORDER FRAME BUNDLE

“…The complete lifting V c of a connection V to L r M C T^M in the sense of A. Morimoto [2] is an example of -4(V).…”

The infinitesimal counterpart of tangent presymplectic groupoids of higher order

“…In the sequel, we adopt the notations of [14] and for the coordinates system U, x i in M , the local coordinates system of T r M over T r U is such that, the coordinate functions x i γ with i = 1, . .…”

“…Tangent lifts of higher order of presymplectic manifolds. In this section we recall briefly the main results of A. Morimoto [14], about the complete lifts of differential forms to the tangent bundle of higher order. These result will be used in the sequel.…”

On natural vector bundle morphisms TA° \otimessq→\otimessq° TAover idTA

“…With such a function f , one can associate the prolongation R); then the dual basis {e * α ; |α| ≤ r} of {e α = j r 0 (z α ); |α| ≤ r} induces the prolongations of functions: f (α) = e * α • T A f , |α| ≤ r (see [6] for T A = T r 1 ).…”