Optical structures, algebraically special spacetimes, and the Goldberg–Sachs theorem in five dimensions

Curvature Invariants in Algebraically Special Spacetimes

Let us define a curvature invariant of the order k as a scalar polynomial constructed from gαβ, the Riemann tensor Rαβγδ, and covariant derivatives of the Riemann tensor up to the order k. According to this definition, the Ricci curvature scalar R or the Kretschmann curvature scalar RαβγδR αβγδ are curvature invariants of the order zero and Rαβγδ;εR αβγδ;ε is a curvature invariant of the order ...

A class of nonfactorisable spacetimes in five dimensions

A one–parameter family of five dimensional nonfactorisable spacetimes is discussed. These spacetimes have Ricci scalar, R = 0. The warp factors, in general, are asymmetric (i.e. different warp factors for the spatial and temporal parts of the 3–brane section), though, with a specific choice of a parameter (denoted here by ν), the asymmetry can be avoided. Over a range of values of ν the energy ...

Generation of the Ahlfors Five Islands Theorem

On algebraically defined graphs in three dimensions

Let F be a field of characteristic zero or of a positive odd characteristic p. For a polynomial f ∈ F[x, y], we define a graph ΓF(xy, f) to be a bipartite graph with vertex partition P ∪L, P = F = L, and (p1, p2, p3) ∈ P is adjacent to [l1, l2, l3] ∈ L if and only if p2 + l2 = p1l1 and p3 + l3 = f(p1, l1). If f = xy, the graph ΓF(xy, xy ) has the length of a shortest cycle (the girth) equal to ...

Weyl’s Theorem for Algebraically Paranormal Operators

Let T be an algebraically paranormal operator acting on Hilbert space. We prove : (i) Weyl’s theorem holds for f(T ) for every f ∈ H(σ(T )); (ii) a-Browder’s theorem holds for f(S) for every S ≺ T and f ∈ H(σ(S)); (iii) the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T . Mathematics Subject Classification (2000). Primary 47A10, 4...