Abstract:
Properties of pairs of product conjugate connections are stated with a special view towards the integrability of the given almost product structure. We define the analogous in product geometry of the structural and the virtual tensors from the Hermitian geometry and express the product conjugate connections in terms of these tensors. Some examples from the geometry of a pair of complementary distributions are discussed and for this case the above structural and virtual tensors are expressed in terms of O'Neill-Gray tensor fields.

Abstract:
Almost para-quaternionic structures on smooth manifolds of dimension 2n are basically equivalent to almost Grassmannian structures of type (2,n). We remind the equivalence and investigate some of its attractive consequences. In particular, we make use of Cartan-geometric techniques in treating the integrability issues on the twistor space level. The discussion includes also the so-called 0-twistor space, which turns out to be the prominent object immediately available both form the para-quaternionic and the Grassmannian point of view.

Abstract:
We discuss two concepts of metric and linear connections in noncommutative geometry, applying them to the case of the product of continuous and discrete (two-point) geometry.

Abstract:
We present an introduction to the geometry of higher-order vector and covector bundles (including higher-order generalizations of the Finsler geometry and Kaluza-Klein gravity) and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by anholonomic frames with associated nonlinear connection structures. We emphasize strong arguments for application of Finsler-like geometries in modern string and gravity theory, noncommutative geometry and noncommutative field theory, and gravity.

Abstract:
We present an introduction to the geometry of higher order vector and co-vector bundles (including higher order generalizations of the Finsler geometry and Kaluza--Klein gravity) and review the basic results on Clifford and spinor structures on spaces with generic local anisotropy modeled by anholonomic frames with associated nonlinear connection structures. We emphasize strong arguments for application of Finsler like geometries in modern string and gravity theory and noncommutative geometry and noncommutative field theory and gravity.

Abstract:
We discuss the interplay between lagrangian distributions and connections in symplectic geometry, beginning with the traditional case of symplectic manifolds and then passing to the more general context of poly- and multisymplectic structures on fiber bundles, which is relevant for the covariant hamiltonian formulation of classical field theory. In particular, we generalize Weinstein's tubular neighborhood theorem for symplectic manifolds carrying a (simple) lagrangian foliation to this situation. In all cases, the Bott connection, or an appropriately extended version thereof, plays a central role.

Abstract:
Using the natural connection equivalent to the SU(2) Yang-Mills instanton on the quaternionic Hopf fibration of $S^7$ over the quaternionic projective space ${\bf HP}^1\simeq S^4$ with an $SU(2)\simeq S^3$ fiber the geometry of entanglement for two qubits is investigated. The relationship between base and fiber i.e. the twisting of the bundle corresponds to the entanglement of the qubits. The measure of entanglement can be related to the length of the shortest geodesic with respect to the Mannoury-Fubini-Study metric on ${\bf HP}^1$ between an arbitrary entangled state, and the separable state nearest to it. Using this result an interpretation of the standard Schmidt decomposition in geometric terms is given. Schmidt states are the nearest and furthest separable ones lying on, or the ones obtained by parallel transport along the geodesic passing through the entangled state. Some examples showing the correspondence between the anolonomy of the connection and entanglement via the geometric phase is shown. Connections with important notions like the Bures-metric, Uhlmann's connection, the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out.

Abstract:
This is the second in a series of papers on natural modification of the normal tractor connection in a parabolic geometry, which naturally prolongs an underlying overdetermined system of invariant differential equations. We give a short review of the general procedure developed in [5] and then compute the prolongation covariant derivatives for a number of interesting examples in projective, conformal and Grassmannian geometries.

Abstract:
We develop the formalism for noncommutative differential geometry and Riemmannian geometry to take full account of the *-algebra structure on the (possibly noncommutative) coordinate ring and the bimodule structure on the differential forms. We show that *-compatible bimodule connections lead to braid operators $\sigma$ in some generality (going beyond the quantum group case) and we develop their role in the exterior algebra. We study metrics in the form of Hermitian structures on Hilbert *-modules and metric compatibility in both the usual and a cotorsion form. We show that the theory works well for the quantum group $C_q[SU_2]$ with its 3D calculus, finding for each point of a 3-parameter space of covariant metrics a unique `Levi-Civita' connection deforming the classical one and characterised by zero torsion, metric-preservation and *-compatibility. Allowing torsion, we find a unique connection with classical limit that is metric-preserving and *-compatible and for which $\sigma$ obeys the braid relations. It projects to a unique `Levi-Civita' connection on the quantum sphere. The theory also works for finite groups and in particular for the permutation group $S_3$ where we find somewhat similar results.