Let M be a connected manifold with a symmetric and complete connection \(\nabla\). Here the authors study the manifolds T(M) and L(M) with respect to the complete lifts of \(\nabla\), giving also some relations between the complete and the vertical lifts of several geometric objects to L(M) and to T(M). By a soul of a noncompact and connected manifold N, the authors understand a connected submanifold S of N such that dim S\(<\dim N\) and the inclusion i: \(S\hookrightarrow N\) is a homotopy equivalence. The results obtained by \textit{T. Higa} in Nagoya Math. J. 96, 41-60 (1984; Zbl 0561.53037) and Comment. Math. Univ. St. Pauli 33, 233-244 (1984; Zbl 0553.53023) are applied here. In the present paper, the spaces of parallel 1-forms and of affine functions on \((L(M),\nabla^ c)\) and on \((T(M),\nabla^ c)\) are determined, then the existence of parallel 1-forms on M is proved to be a sufficient condition to obtain souls for \((T(M),\nabla^ c)\) and when this is possible, some relations between the above souls and the corresponding souls of (M,\(\nabla)\) are given. At the end, some reducibility theorems for T(M) are stated.