Czechoslovak Mathematical Journal, Vol. 67, No. 2, pp. 297-316, 2017


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

Jiří Tomáš

Received October 23, 2015.  First published March 1, 2017.

Abstract:  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 $m$. 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 $J^r(M,N)$.
Keywords:  $r$-jet; bundle functor; Weil functor; Lie group; jet group; $B$-admissible $A$-velocity
Classification MSC:  58A20, 58A32, 53C24
DOI:  10.21136/CMJ.2017.0566-15


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Affiliations:   Jiří Tomáš, Department of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Technická 2, Brno, Czech Republic, e-mail: Tomas@fme.vutbr.cz

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