The author reviews the theory of approximate infinitesimal symmetries of partial differential equations. Based on this and on Ibragimov's result on the general symmetries of the vacuum Einstein equation, he proposes a method to calculate approximate symmetries of the non-vacuum Einstein equation: the energy-momentum tensor is treated like a perturbation.

By taking into account the work of J. Rataj and M. Zähle [Geom. Dedicata 57, 259-283 (1995; Zbl 0844.53050)], R. Schneider and W. Weil [Math. Nachr. 129, 67-80 (1986; Zbl 0602.52003)], W. Weil [Math. Z. 205, 531-549 (1990; Zbl 0705.52006)], an integral formula is obtained here by using the technique of rectifiable currents.This is an iterated version of the principal kinematic formula for $q$ sets of positive reach and generalized curvature measures.

[For the entire collection see Zbl 0742.00067.]The Penrose transform is always based on a diagram of homogeneous spaces. Here the case corresponding to the orthogonal group $SO(2n,C)$ is studied by means of Clifford analysis [see F. Brackx, R. Delanghe and F. Sommen: Clifford analysis (1982; Zbl 0529.30001)], and is presented a simple approach using the Dolbeault realization of the corresponding cohomology groups and a simple calculus with differential forms (the Cauchy integral formula for solutions of...

[For the entire collection see Zbl 0742.00067.]Let $P^m$ be the set of hyperplanes $\sigma :\langle \overrightarrow{x},\overrightarrow{\theta }\rangle =p$ in ${ℛ}^m$, $S^{m-1}$ the unit sphere of ${ℛ}^m$, $E^m$ the exterior of the unit ball, $T^m$ the set of hyperplanes not passing through the unit ball, $Rf(\overrightarrow{\theta },p)={\int }_{\sigma }f\left(\overrightarrow{x}\right)d\overrightarrow{x}$ the Radon transform, $R^{\#}g\left(\overrightarrow{x}\right)={\int }_{S^{m-1}}g(\overrightarrow{\theta },\langle \overrightarrow{x},\overrightarrow{\theta }\rangle )d{S}_{\overrightarrow{\theta }}$ its dual. $R$ as operator from $L^2\left({ℛ}^m\right)$ to $L^2(S^{m-1)}\times ℛ)$ is a closable, densely defined operator, $R^{*}$ denotes the operator given by $\left(R^{*}g\right)\left(\overrightarrow{x}\right)=R^{\#}g\left(\overrightarrow{x}\right)$ if the integral exists for $\overrightarrow{x}\in {ℛ}^m$ a.e. Then the closure of $R^{*}$ is the adjoint of $R$. The author shows that the Radon transform and its dual can be linked by two operators...

[For the entire collection see Zbl 0742.00067.]Differential spaces, whose theory was initiated by R. Sikorski in the sixties, provide an abstract setting for differential geometry. In this paper the author studies the wedge sum of such spaces and deduces some basic results concerning this construction.

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 ${ℛ}^{2l},S^{2l},{\mathscr{H}}^{2l}$.