The natural operators $T^{(0,0)}\rightsquigarrow T^{(1,1)}T^{(r)}$

Włodzimierz M. Mikulski¹

¹ Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland

Colloquium Mathematicum, Tome 96 (2003) no. 1, pp. 5-16.

Résumé

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

Détail

DOI : 10.4064/cm96-1-2

Keywords: study problem map mathbb n manifold induces canonically affinor vector r tangent bundle mathbb * problem reflected concept natural operators cal rightsquigarrow integers geq geq prove space operators dimensional module cal infty mathbb construct explicitly basis module par