The natural linear operators T*→TT(r)

Abstract

For natural numbers n 3 and r a complete description of all natural bilinear operators T Mfn T (0;0) T (0;0) T (r) is presented. Next for natural numbers r and n 3 a full classication of all natural linear operators T jMfn TT (r) is obtained. Introduction. Letn andr be natural numbers. Given ann-dimensional manifold M we have the r-tangent vector bundle T (r) M = (J r (M;R)0) over M. Every embedding ' : M ! N of n-manifolds induces a vector bundle mapT (r) ' : T (r) M! T (r) N covering' such thathT (r) '(!);j r '(x) i =h!;j r x( ')i for !2 T (r) x M, j r '(x) 2 J r '(x) (N;R)0, x2 M. The corre- spondence T (r) :Mfn!FM is a bundle functor from the categoryMfn of n-manifolds and embeddings into the categoryFM of b ered manifolds and b ered maps (3). In (4), we studied the problem of how a 1-form ! 2 1 (M) on an n- manifold M can induce a 1-form A(!)2 1 (T (r) M) on T (r) M. This prob- lem was reected in the concept of natural linear operators T jMfn T T (r) in the sense of Kol a r, Michor and Slov ak (3). We presented a complete de- scription of such operators. In the present note we start with the problem of how a 1-form ! 2 1 (M) and a map f : M ! R on an n-manifold M can induce a map B(!;f) : T (r) M ! R. This problem concerns natural bilinear operators B : T Mfn T (0;0) T (0;0) T (r) . We prove that the vector space of all such operators is 0-dimensional if n 3 and r 3, 3-dimensional if n 3 and r = 2, and 2-dimensional if n 3 and r = 1. We construct explicit bases of the vector spaces in question. Next, using this classication we investigate how a 1-form ! on an n- manifold M can induce a vector eld C(!) on T (r) M. This problem relates to natural linear operators C : T jMfn TT (r) . We deduce that the vector space of all such operators is 0-dimensional ifn 3 andr 3, 2-dimensional