Summary: We specialize in a new way the second Noether theorem for gauge-natural field theories by relating it to the Jacobi morphism and show that it plays a fundamental role in the derivation of canonical covariant conserved quantities. In particular we show that Bergmann-Bianchi identities for such theories hold true covariantly and canonically only along solutions of generalized gauge-natural Jacobi equations. Vice versa, all vertical parts of gauge-natural lifts of infinitesimal principal automorphisms...

The author considers the Nijenhuis map assigning to two type (1,1) tensor fields $\alpha$, $\beta$ a mapping $$\langle \alpha ,\beta \rangle :(\xi ,\zeta )\mapsto \left[\alpha \right(\xi ),\beta (\zeta \left)\right]+\alpha \circ \beta \left(\right[\xi ,\zeta \left]\right)-\alpha \left(\right[\xi ,\beta \left(\zeta \right)\left]\right)-\beta \left(\right[\alpha \left(\xi \right),\zeta \left)\right]),$$ where $\xi$, $\zeta$ are vector fields. Then $\langle \alpha ,\beta \rangle$ is a type (2,1) tensor field (Nijenhuis tensor) if and only if $[\alpha ,\beta ]=0$. Considering a smooth manifold $X$ with a smooth action of a Lie group, a secondary invariant may be defined as a mapping whose area of invariance is restricted to the inverse image of an invariant subset of $X$ under another invariant mapping. The author recognizes a secondary invariant related to the...

Summary: [For the entire collection see Zbl 0742.00067.]A general theory of fibre bundles structured by an arbitrary differential-geometric category is presented. It is proved that the structured bundles of finite type coincide with the classical associated bundles.

[For the entire collection see Zbl 0699.00032.] A manifold (M,g) is said to be generalized Einstein manifold if the following condition is satisfied $$\left({\nabla }_{X}S\right)(Y,Z)=\sigma \left(X\right)g(Y,Z)+\nu \left(Y\right)g(X,Z)+\nu \left(Z\right)g(X,Y)$$ where S(X,Y) is the Ricci tensor of (M,g) and $\sigma$ (X), $\nu$ (X) are certain $\ell$-forms. In the present paper the author studies properties of conformal and geodesic mappings of generalized Einstein manifolds. He gives the local classification of generalized Einstein manifolds when g($\psi$ (X),$\psi$ (X))$\ne 0$.

Summary: We provide a geometric interpretation of generalized Jacobi morphisms in the framework of finite order variational sequences. Jacobi morphisms arise classically as an outcome of an invariant decomposition of the second variation of a Lagrangian. Here they are characterized in the context of generalized Lagrangian symmetries in terms of variational Lie derivatives of generalized Euler-Lagrange morphisms. We introduce the variational vertical derivative and stress its link with the classical...

Summary: Geodesics and curvature of semidirect product groups with right invariant metrics are determined. In the special case of an isometric semidirect product, the curvature is shown to be the sum of the curvature of the two groups. A series of examples, like the magnetic extension of a group, are then considered.

[For the entire collection see Zbl 0742.00067.]Let $G$ be a connected semisimple Lie group with finite center. In this review article the author describes first the geometric realization of the discrete series representations of $G$ on Dolbeault cohomology spaces and the tempered series of representations of $G$ on partial Dolbeault cohomology spaces. Then he discusses his joint work with Wilfried Schmid on the construction of maximal globalizations of standard Zuckerman modules via geometric quantization....