a gyrogroup is a nonassociative group-like structure modelled on the space of relativistically admissible velocities with a binary operation given by einstein's velocity addition law. in this article, we present a few of groups sitting inside a gyrogroup $g$, including the commutator subgyrogroup, the left nucleus, and the radical of $g$. the normal closure of the commutator subgyrogroup, ...

If M is the underlying smooth oriented 4-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics h on M such that W(ω, ω) > 0, where W is the self-dual Weyl curvature of h, and ω is a non-trivial self-dual harmonic 2-form on (M,h). While this open region in the space of Riemannian metrics contains all the known Einstein metrics on M , we show that it contains no others. Conse...

Let (Mn, g), n ≥ 4, be a compact simply-connected Riemannian manifold with nonnegative isotropic curvature. Given 0 < l ≤ L, we prove that there exists ε = ε(l, L, n) satisfying the following: If the scalar curvature s of g satisfies l ≤ s ≤ L and the Einstein tensor satisfies |Ric − s n g| ≤ ε then M is diffeomorphic to a symmetric space of compact type. This is a smooth analogue of the result...

The structure of the general, inhomogeneous solution of (bosonic) Einstein-matter systems in the vicinity of a cosmological singularity is considered. We review the proof (based on ideas of BelinskiiKhalatnikov-Lifshitz and technically simplified by the use of the ArnowittDeser-Misner Hamiltonian formalism) that the asymptotic behaviour, as one approaches the singularity, of the general solutio...

Wormholes, Time machines and Energy conditions: would Einstein have come to this parallel session? If it was held in 1950, I don’t think so. Einstein had a strong belief in preserving causality as a fundamental principle that contributed to his work on relativity. But if Einstein was at MG11 in 2006, this is where Einstein would be. Einstein would be attracted not so much for his interest in ge...

Warped products provide a rich class of physically significant geometric objects. Warped product construction is an important method to produce a new metric with a base manifold and a fibre. We construct compact base manifolds with a positive scalar curvature which do not admit any non-trivial quasi-Einstein warped product, and non compact complete base manifolds which do not admit any non-triv...