Natural affinors on \((J^{r,s,q}(\cdot , R^{1,1})_0)^{\ast }\). (English) Zbl 1090.58501

Mikulski, Włodzimierz M.

Commentat. Math. Univ. Carol. 42, No. 4, 655-663 (2001).

Consider fibered manifolds \(Y \to M\) and \(Z \to N\), and fibered morphisms \(f,g: Y \to Z\) with base maps \(\underline {f}, \underline {g}: M \to N\). Let \(r,s,q\) be natural numbers such that \(s \geq r \leq q\). The \(f,g\) are said to determine the same \((r,s,q)\)-jet \(j_y^{r,s,q}f = j_y^{r,s,q}g\) at \(y \in Y_x\), \(x \in M\), if \(j^r_yf = j^r_yg\), \(j^s_y(f| Y_x) = j^s_y(g| Y_x)\) and \(j^q_x \underline {f} = j^q_x \underline {g}\). The space of all \((r,s,q)\)-jets of \(Y\) into \(Z\) is denoted by \(J^{r,s,q}(Y,Z)\). The paper deals with the special case \(Z = R^{1,1}\), the trivial bundle \(R \times R \to R\). In this case, the space \(J^{r,s,q}(Y,R^{1,1})_0\), \(0 \in R^2\), has an induced structure of a vector bundle over \(Y\); the dual vector bundle is denoted by \((J^{r,s,q}(Y,R^{1,1})_0)^*\). The paper is clearly written. The results concern the classification of
(1) All natural transformations \((J^{r,s,q}(Y, R^{1,1})_0)^* \to (J^{r,s,q}(Y,R^{1,1})_0)^*\); this provides an extension of I.Kolář and G.Vosmanská [Cas. Pestovani Mat. 114, 181–186 (1989; Zbl 0678.58002)].
(2) All linear natural transformations \(T(J^{r,s,q}(Y, R^{1,1})_0)^* \to (J^{r,s,q}(Y,R^{1,1})_0)^*\).
(3) All natural transformations \(T(J^{r,s,q}(Y, R^{1,1})_0)^* \to TY\).
(4) All natural affinors on \((J^{r,s,q}(Y, R^{1,1})_0)^*\). It is proved that the set of all natural affinors is a \(3\)-dimensional vector space over \(R\), and its basis is explicitly constructed. As a consequence, it appears that there is no natural generalized connection on \((J^{r,s,q}(Y, R^{1,1})_0)^*\).
Similar problems are solved for the case of \((J^{r,s}(Y, R)_0)^*\) instead of \((J^{r,s,q}(Y, R^{1,1})_0)^*\).

Reviewer: Olga Krupková (Opava)

MSC:

58A20	Jets in global analysis
53A55	Differential invariants (local theory), geometric objects

Keywords:

bundle functors; natural transformations; natural affinors

Citations:

Zbl 0678.58002