The natural operators
Tom 96 / 2003
Colloquium Mathematicum 96 (2003), 5-16
MSC: 58A20, 53A55.
DOI: 10.4064/cm96-1-2
Streszczenie
We study the problem of how a map f:M\to {{\mathbb R}} on an n-manifold M induces canonically an affinor A(f):TT^{(r)}M\to TT^{(r)}M on the vector r-tangent bundle T^{(r)}M=(J^r(M,{{\mathbb R}})_0)^* over M. This problem is reflected in the concept of natural operators A:T^{(0,0)}_{| {\cal M} f_n} \rightsquigarrow T^{(1,1)}T^{(r)}. For integers r\geq 1 and n\geq 2 we prove that the space of all such operators is a free (r+1)^2-dimensional module over {\cal C}^\infty (T^{(r)}{{\mathbb R}}) and we construct explicitly a basis of this module. \par