The torsions of a general connection $\Gamma$ on the $r$th-order tangent bundle of a manifold $M$ are defined as the Frölicher-Nijenhuis bracket of $\Gamma$ with the natural affinors. The author deduces the basic properties of these torsions. Then he compares them with the classical torsion of a principal connection on the $r$th-order frame bundle of $M$.

[For the entire collection see Zbl 0699.00032.] A fibration $F\to E\to B$ is called totally noncohomologuous to zero (TNCZ) with respect to the coefficient field k, if $H^{*}(E;k)\to H^{*}(F;k)$ is surjective. This is equivalent to saying that ${\pi }_1\left(B\right)$ acts trivially on $H^{*}(F;k)$ and the Serre spectral sequence collapses at $E^2$. S. Halperin conjectured that for $char\left(k\right)=0$ and F a 1-connected rationally elliptic space (i.e., both $H^{*}(F;𝒬)$ and ${\pi }_{*}\left(F\right)\otimes 𝒬$ are finite dimensional) such that $H^{*}(F;k)$ vanishes in odd degrees, every fibration $F\to E\to B$ is TNCZ. The author proves this being the case...

Des liens inattendus ont été récemment mis à jour entre le transport optimal de Monge–Kantorovich et certains problèmes de géométrie riemannienne, en liaison avec la courbure de Ricci. Une des retombées de ces interactions est la naissance d’une théorie « synthétique » des espaces métriques mesurés à courbure de Ricci minorée, venant compléter la théorie classique des espaces métriques à courbure sectionnelle minorée. Dans ce texte (également fourni aux actes du Séminaire d’Équations aux dérivées...

For the entire collection see Zbl 0699.00032.

Summary: We describe explicitly the kernels of higher spin twistor operators on standard even dimensional Euclidean space ${ℛ}^{2l}$, standard even dimensional sphere $S^{2l}$, and standard even dimensional hyperbolic space ${\mathscr{H}}^{2l}$, using realizations of invariant differential operators inside spinor valued differential forms. The kernels are finite dimensional vector spaces (of the same cardinality) generated by spinor valued polynomials on ${ℛ}^{2l},S^{2l},{\mathscr{H}}^{2l}$.

[For the entire collection see Zbl 0699.00032.] In a previous paper [Cas. Pestovani Mat. 115, No.4, 360-367 (1990)] the author determined the set of the vector fields on TM by which connections on TM can be constructed. In this paper, he generalizes some of such constructions to the case of vector fields on fibred manifolds, giving several examples.

Let $X$ be the interior of a compact manifold $\overline{X}$ of dimension $n+1$ with boundary $M=\partial X$, and $g_{+}$ be a conformally compact metric on $X$, namely $\overline{g}\equiv r^2{g}_{+}$ extends continuously (or with some degree of smoothness) as a metric to $X$, where $r$ denotes a defining function for $M$, i.e. $r>0$ on $X$ and $r=0$, $dr\ne 0$ on $M$. The restrction of $\overline{g}$ to $TM$ rescales upon changing $r$, so defines invariantly a conformal class of metrics on $M$, which is called the conformal infinity of $g_{+}$. In the present paper, the author considers conformally compact metrics...

In the paper the origins of the intrinsic unitary symmetry encountered in the study of bosonic systems with finite degrees of freedom and its relations with the Weyl algebra (1979, Jacobson) generated by the quantum canonical commutation relations are presented. An analytical representation of the Weyl algebra formulated in terms of partial differential operators with polynomial coefficients is studied in detail. As a basic example, the symmetry properties of the $d$-dimensional quantum harmonic oscillator...