Geometric realizations, curvature decompositions, and Weyl manifolds

Abstract

We show any Weyl curvature model can be geometrically realized by a Weyl manifol

Curvature Properties of Weyl Geometries

“…We have the following result (see, for example, the discussion in [5,11] R(x, y, z, w) + R(x, y, w, z) = 0.…”

“…The following result (see, for example, Theorem 6 [5]) can be used to construct Weyl manifolds: (N, g). Let φ be a smooth 1-form on N .…”

“…Suppose that Assertion (4) holds. We apply Theorem 6 of [5] to see dφ = − 1 n Λ Ric. Since the curvature tensor is algebraic, Λ Ric = 0.…”

Acceleration in Weyl Integrable Spacetime

“…where the 1-form Q µ is customarily called Weyl covariant vector field (see for example [15,16] and the Appendix in [11] for a detailed exposition of the techniques involved in Weyl geometry). We denote by D the Levi-Civita connection of the metric g αβ .…”

Geometric Realizations of Affine Kähler Curvature Models

“…Such a structure is said to be geometrically realizable by a Weyl manifold if there exists a Weyl manifold (M, g, ∇), a point P ∈ M , and an isomorphism Ξ from V to T P M so that Ξ * g P = ·, · and so that Ξ * R P = A. The following result [7] …”