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Let F be a codimension–one, C 2-foliation on a manifold M without boundary. In this work we show that if the Godbillon–Vey class GV (F) ∈ H 3 (M) is non-zero, then F has a hyperbolic resilient leaf. Our approach is based on methods of C 1-dynamical systems, and does not use the classification theory of C 2-foliations. We first prove that for a codi...
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In De Sitter / Anti De Sitter space-time and in other geometries, reference sub-manifolds from which proper time is measured along integral curves, are described as events. We introduce here a foliation with the help of a scalar field. The scalar field need not be unique but from the gradient of the scalar field, an intrinsic Reeb vector of the foliations perpendicular to the gradient vector is calculated. The Reeb vector describes the acceleration of a physical particle that moves along the integral curves that are formed by the gradient of the scalar field. The Reeb vector appears as a component of an anti-symmetric matrix which is a part of a rank-2, 2-Form. The 2-form is extended into a non-degenerate 4-form and into rank-4 matrix of a 2-form, which when multiplied by a velocity of a particle, becomes the acceleration of the particle. The matrix has one U(1) degree of freedom and an additional SU(2) degrees of freedom in two vectors that span the plane perpendicular to the gradient of the scalar field and to the Reeb vector. In total, there are U(1) x SU(2) degrees of freedom. SU(3) degrees of freedom arise from three dimensional foliations but require an additional symmetry to exist in order to have a valid covariant meaning.
Matter in the Einstein Grossmann equation is replaced by the action of the acceleration field, i.e. by a geometric action which is not anticipated by the metric alone. This idea leads to a new formalism that replaces the conventional stress-energy-momentum-tensor. The formalism will be mainly developed for classical physics but will also be discussed for quantized physics based on events instead of particles. The result is that a positive charge manifests small attracting gravity and a stronger but small repelling acceleration field that repels even uncharged particles that have a rest mass. Negative charge manifests a repelling anti-gravity but also a stronger acceleration field that attracts even uncharged particles that have rest mass.
Preliminary version: http://sciencedomain.org/abstract/9858
