Plane Collineations

Proper Matter Collineations of Plane Symmetric Spacetimes

Let (M, g) be a spacetime, where M is a smooth, connected, Hausdorff four-dimensional manifold and g is smooth Lorentzian metric of signature (+ -) defined on M . The manifold M and the metric g are assumed smooth (C). A smooth vector field ξ is said to preserve a matter symmetry [1] on M if, for each smooth local diffeomorphism φt associated with ξ, the tensors T and φ∗tT are equal on the doma...

Weyl collineations that are not curvature collineations

Though the Weyl tensor is a linear combination of the curvature tensor, Ricci tensor and Ricci scalar, it does not have all and only the Lie symmetries of these tensors since it is possible, in principle, that “asymmetries cancel”. Here we investigate if, when and how the symmetries can be different. It is found that we can obtain a metric with a finite dimensional Lie algebra of Weyl symmetrie...

Complete collineations revisited

This paper takes a new look at some old spaces. The old spaces are the moduli spaces of complete collineations, introduced and explored by many of the leading lights of 19th-century algebraic geometry, such as Chasles, Schubert, Hirst, and Giambelli. They are roughly compactifications of the spaces of linear maps of a fixed rank between two fixed vector spaces, in which the boundary added is a ...

Collineations in Generalized Spaces

Classification of Static Plane Symmetric Spacetimes according to their Matter Collineations

In this paper we classify static plane symmetric spacetimes according to their matter collineations. These have been studied for both cases when the energy-momentum tensor is non-degenerate and also when it is degenerate. It turns out that the non-degenerate case yields either four, five, six, seven or ten independent matter collineations in which four are isometries and the rest are proper. Th...