In this paper, we give a conceptual explanation of dark energy as a small negative residual scalar curvature present even in empty spacetime. This curvature ultimately results from postulating a discrete spacetime geometry, very closely related to that used in the dynamical triangulations approach to quantum gravity. In this model, there are no states which have total scalar curvature exactly z...

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...

In this paper we prove two theorems on initial data for general relativity. The results present “rigidity phenomena” for the extrinsic curvature, caused by the negativity of the scalar curvature. More precisely, Theorem 1 states that in the case of non-vacuum initial data if the metric has everywhere non-positive scalar curvature then the extrinsic curvature cannot be compactly supported. Theor...

We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact be realized by a Hermitian manifold of constant scalar curvature and constant ⋆-scalar curvature which satisfies the Kaehler condition at the point in question.

In this note we prove a theorem on non-vacuum initial data for general relativity. The result presents a “rigidity phenomenon” for the extrinsic curvature, caused by the nonpositive scalar curvature. More precisely, we state that in the case of asymptotically flat non-vacuum initial data if the metric has everywhere non-positive scalar curvature then the extrinsic curvature cannot be compactly ...

We obtain the evolution equations for the Riemann tensor, the Ricci tensor and the scalar curvature induced by the mean curvature flow. The evolution for the scalar curvature is similar to the Ricci flow, however, negative, rather than positive, curvature is preserved. Our results are valid in any dimension.

Hitchin proved that if M is a spin manifold with positive scalar curvature, then the A^O-characteristic number a(M) vanishes. Gromov and Lawson conjectured that for a simply connected spin manifold M of dimension > 5, the vanishing of a(M) is sufficient for the existence of a Riemannian metric on M with positive scalar curvature. We prove this conjecture using techniques from stable homotopy th...