On the Geometry of Vertical Weil Bundles

We describe some general geometric properties of the fiber product preserving bundle functors. Special attention is paid to the vertical Weil bundles. We discuss namely the flow natural maps and the functorial prolongation of connections. The main purpose of the present paper is to describe some geometric properties of the category FMm of fibered manifolds with m-dimensional bases and local fibered morhisms with local diffeomorphisms as base maps. Special attention is paid to the vertical Weil functors V . In Section 1 we present the covariant approach to the Weil functors on the category Mf of all smooth manifolds and all smooth maps. We mention the fundamental theoretical result that the classical Weil functors T coincide with the product preserving bundle functors onMf . In Section 2 we introduce the Weil fields as the sections of Weil bundles and we describe their basic properties. Section 3 is devoted to the concept of flow natural map, that represents a suitable tool for constructing the flow prolongation of a projectable vector field on a fibered manifold Y →M . The last section describes the functorial prolongation of connections with respect to a fiber product preserving bundle functor on FMm. Unless otherwise specified, we use the terminology and notation from [6]. All manifolds and maps are assumed to be infinitely differentiable. 1. Fiber product preserving bundle functors We recall that a Weil algebra is a finite dimensional, commutative, associative and unital algebra of the form A = R×N , where N is the ideal of all nilpotent elements of A. There exists an integer r such that Nr+1 = 0, the smallest r with this property is called the order of A. On the other hand, the dimension wA of the vector space N/N2 is the width of A. We say that a Weil algebra of width k and order r is a Weil (k, r)-algebra, [5]. The simpliest example of a Weil algebra is Dk = R[x1, . . . , xk] / 〈x1, . . . , xk〉 = J 0 (R,R) . 2010 Mathematics Subject Classification: primary 58A20; secondary 58A32, 53C05.