We solve the Ricci-flat equations of extended general relativity to obtain an interesting class of cosmological models. The solutions are analogous to the 4D ones of Bianchi type-I of Kasner type and have significant implications for astrophysics.

We prove that, starting at an initial metric g(0) = e2u0(dx2 + dy2) on R2 with bounded scalar curvature and bounded u0, the Ricci flow ∂tg(t) = −Rg(t)g(t) converges to a flat metric on R2.

We study G-invariant Kähler metrics on G from the Hamiltonian point of view. As an application we show that there exist G × G-invariant Ricci-flat Kähler metrics on G for any compact semisimple Lie group G.

It is a classical fact that the cotangent bundle T M of a differentiable manifold M enjoys a canonical symplectic form Ω. If (M, J, g, ω) is a pseudo-Kähler or para-Kähler 2n-dimensional manifold, we prove that the tangent bundle TM also enjoys a natural pseudo-Kähler or para-Kähler structure (J̃, g̃,Ω), where Ω is the pull-back by g of Ω and g̃ is a pseudoRiemannian metric with neutral signature ...

For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal tractor holonomy as affine holonomy of such a connection. We give an example of an ambient connectio...

For a conformal manifold we introduce the notion of an ambient connection, an affine connection on an ambient manifold of the conformal manifold, possibly with torsion, and with conditions relating it to the conformal structure. The purpose of this construction is to realise the normal conformal tractor holonomy as affine holonomy of such a connection. We give an example of an ambient connectio...

The D-dimensional cosmological model on the manifold M = R × M1 × M2 describing the evolution of 2 Einsteinian factor spaces, M1 and M2, in the presence of multicomponent perfect fluid source is considered. The barotropic equation of state for mass-energy densities and the pressures of the components is assumed in each space. When the number of the non Ricci-flat factor spaces and the number of...

Under intrinsic and extrinsic curvature assumptions on a Riemannian spin manifold and its boundary, we show that there is an isomorphism between the restriction to the boundary of parallel spinors and extrinsic Killing spinors of nonnegative Killing constant. As a corollary, we prove that a complete Ricci-flat spin manifold with mean-convex boundary isometric to a round sphere, is necessarily a...