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Let F : FM → VB be a vector bundle functor. First we classify all natural operators Tproj|FMm,n T (F|FMm,n )∗ transforming projectable vector fields on Y to functions on the dual bundle (FY )∗ for any FMm,n-object Y . Next, under some assumption on F we study natural operators T ∗ hor|FMm,n T (F|FMm,n )∗ lifting horizontal 1-forms on Y to 1-forms on (FY )∗ for any Y as above. As an application we classify natural operators T ∗ hor|FMm,n T (F|FMm,n )∗ for some vector bundle functors F on fibered manifolds. 0. Introduction. In this paper we consider the following categories over manifolds: the categoryMf of manifolds and maps, the categoryMfm of m-dimensional manifolds and embeddings, the category FM of fibered manifolds and fibered maps, the category FMm,n of fibered manifolds with m-dimensional bases and n-dimensional fibers and fibered embeddings, and the category VB of all vector bundles and vector bundle maps. The notions of bundle functors and natural operators can be found in the fundamental monograph [3]. In [5], given a vector bundle functor F : Mf → VB we classified all natural operators A : T|Mfm T (F|Mfm) transforming a vector field X on an m-manifold M into a function A(X) : (FM)∗ → R on the dual vector bundle (FM)∗ and proved that every natural operator B : T ∗ |Mfm T (F|Mfm) ∗ transforming a 1-form ω on an m-manifold M into a 1-form B(ω) on (FM)∗ is of the form B(ω) = aω +λ for some uniquely determined canonical map a : (FM)∗ → R and some canonical 1-form λ on (FM)∗. These results were generalizations of [1, 4]. In the present paper we study similar problems for a vector bundle functor F : FM → VB on a fibered manifold instead of on a manifold. Modifying methods from [5], for natural numbers m and n we classify all natural operators A : Tproj|FMm,n T (F|FMm,n) transform2000 Mathematics Subject Classification: Primary 58A20.

The natural operators lifting horizontal 1-forms to some vector bundle functors on fibered manifolds