In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This gives a connection between links and homotopy groups. Similar results hold for br...

We introduce a new spinorial, BF-like action for the Einstein gravity. This is a first, up to our knowledge, 2-form action which describes the real, Lorentzian gravity and uses only the self-dual connection. In the generic case, the corresponding classical canonical theory is equivalent to the Einstein-Ashtekar theory plus the reality conditions. [email protected] [email protected]

We point out a connection between reflection principles and generic large cardinals. One principle of pure reflection is introduced that is as strong as generic supercompactness of ω2 by σ-closed forcing. This new concept implies CH and extends the reflection principles for stationary sets in a canonical way.

The Hamiltonian form of the Hilbert action in the first order tetrad formalism is examined. We perform a non-linear field redefinition of the canonical variables isolating the part of the spin connection which is canonically conjugate to the tetrad. The geometrical meaning of the constraints written in these new variables is examined.

Abstract An intrinsic geometric decomposition is applied to provide via a canonical third-order nonlinear equation connection between 2+1-dimensional fibre-reinforced motions of fluid and magnetohydrodynamic system. A Lagrange-type parametrisation introduced, whereby both algebraic properties certain non-steady may be established.

We show that the classical Hamburger moment problem can be included in the spectral theory of generalized indefinite strings. Namely, we introduce the class of Krein–Langer strings and show that there is a bijective correspondence between moment sequences and this class of generalized indefinite strings. This result can be viewed as a complement to the classical results of M. G. Krein on the co...

After a brief chronological sketch of developments in non-perturbative canonical quantum gravity, some of the recent mathematical results are reviewed. These include: i) an explicit construction of the quantum counterpart of Wheeler’s superspace; ii) a rigorous procedure leading to the general solution of the diffeomorphism constraint in quantum geometrodynamics as well as connection dynamics; ...

Our purpose is to study a space ? of centered m-planes in n-projective space. Generalized fiberings (with semi-gluing) are investigated. Planar and normal affine connections associated with the set generalized fiberings. Fields these connection objects define torsion curvature tensors. The canonical cases planar considered.

Let V be a holomorphic vector bundle on a complex manifold M , with a flat connection ∇. We shall make the following three assumptions: (1) M is an open subset of a bigger complex manifold M . (2) The boundary D = M \M is a divisor with normal crossing singularities. (3) The connection ∇ is unipotent along D. Under these assumptions, V has a canonical extension to a vector bundle V on M . Since...

A geometric quantization of a Kähler manifold, viewed as a symplectic manifold, depends on the complex structure compatible with the symplectic form. The quantizations form a vector bundle over the space of such complex structures. Having a canonical quantization would amount to finding a natural (projectively) flat connection on this vector bundle. We prove that for a broad class of manifolds,...