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.

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 $M$.

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