Projectable nonlinear connections.
Projective Finsler connections. (Connections Finsler-projectives.)
Prolongation of projectable tangent valued forms
First we deduce some general properties of product preserving bundle functors on the category of fibered manifolds. Then we study the prolongation of projectable tangent valued forms with respect to these functors and describe the complete lift of the Frölicher-Nijenhuis bracket. We also present the coordinate formula for composition of semiholonomic jets.
Prolongation of second order connections to vertical Weil bundles
We study systematically the prolongation of second order connections in the sense of C. Ehresmann from a fibered manifold into its vertical bundle determined by a Weil algebra . In certain situations we deduce new properties of the prolongation of first order connections. Our original tool is a general concept of a -field for another Weil algebra and of its -prolongation.
Prolongation of tangent valued forms to Weil bundles
We prove that the so-called complete lifting of tangent valued forms from a manifold to an arbitrary Weil bundle over preserves the Frölicher-Nijenhuis bracket. We also deduce that the complete lifts of connections are torsion-free in the sense of M. Modugno and the second author.
Pseudo-Riemannian structures with the same connection.
Quantum principal bundles and their characteristic classes
A general theory of characteristic classes of quantum principal bundles is presented, incorporating basic ideas of classical Weil theory into the conceptual framework of noncommutative differential geometry. A purely cohomological interpretation of the Weil homomorphism is given, together with a geometrical interpretation via quantum invariant polynomials. A natural spectral sequence is described. Some interesting quantum phenomena appearing in the formalism are discussed.
Quaternion Structure on the Moduli Space of Yang-Mills Connections.
Real projective structures on Riemann surfaces
Reduction theorem for general connections
We prove the (first) reduction theorem for general and classical connections, i.e. we prove that any natural operator of a general connection Γ on a fibered manifold and a classical connection Λ on the base manifold can be expressed as a zero order operator of the curvature tensors of Γ and Λ and their appropriate derivatives.
Refined Kato inequalities in riemannian geometry
We describe the recent joint work of the author with David M. J. Calderbank and Paul Gauduchon on refined Kato inequalities for sections of vector bundles living in the kernel of natural first-order elliptic operators.
Regular infinite dimensional Lie groups.
Relations between linear connections on the tangent bundle and connections on the jet bundle of a fibred manifold.
Relations between linear connections on the tangent bundle and connections on the jet bundle of a fibred manifold
All natural operations transforming linear connections on the tangent bundle of a fibred manifold to connections on the 1-jet bundle are classified. It is proved that such operators form a 2-parameter family (with real coefficients).
Relèvement horizontal des connexions linéaires au fibré vectoriel associé avec le fibré principal des repères linéaires
Relèvements horizontaux de tenseurs de type (1,1) au fibré E = TM⊗T* M
Remarks on Grassmannian Symmetric Spaces
The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for -graded parabolic geometries and for almost Grassmannian structures, in particular. As an application of two general constructions with parabolic geometries, we present an example of non-flat Grassmannian symmetric space. Next we observe there is a distinguished torsion-free...
Remarks on symmetries of parabolic geometries
We consider symmetries on filtered manifolds and we study the -graded parabolic geometries in more details. We discuss the existence of symmetries on the homogeneous models and we conclude some simple observations on the general curved geometries.
Remarks on the Nijenhuis tensor and almost complex connections
Remarques sur les systèmes différentiels