The general rigidity result for bundles of $A$-covelocities and $A$-jets

Tomáš, Jiří

Czechoslovak Mathematical Journal, Tome 67 (2017) no. 2, pp. 297-316.

Résumé

Let $M$ be an $m$-dimensional manifold and $A=\mathbb D^r_k /I=\mathbb R \oplus N_A$ a Weil algebra of height $r$. We prove that any $A$-covelocity $T^A_x f \in T^{A*}_x M$, $x \in M$ is determined by its values over arbitrary $\max \{\mathop {\rm width}A, m \}$ regular and under the first jet projection linearly independent elements of $T^A_xM$. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result $T^{A*}M \simeq T^{r*}M$ without coordinate computations, which improves and generalizes the partial result obtained in Tomáš (2009) from $m \ge k$ to all cases of $m$. \endgraf We also introduce the space $J^A(M,N)$ of $A$-jets and prove its rigidity in the sense of its coincidence with the classical jet space $J^r(M,N)$.

Détail

DOI : 10.21136/CMJ.2017.0566-15

Classification : 53C24, 58A20, 58A32
Keywords: $r$-jet; bundle functor; Weil functor; Lie group; jet group; $B$-admissible $A$-velocity