We study an even dimensional manifold with a pseudo-Riemannian metric with arbitrary signature and arbitrary dimensions. We consider the Ricci flat equations and present a procedure to construct solutions to some higher (even) dimensional Ricci flat field equations from the four dimensional Ricci flat metrics. When the four dimensional Ricci flat geometry corresponds to a colliding gravitationa...

Using very weak criteria for what may constitute a noncommutative geometry, I show that a pseudo-Riemannian manifold can only be smoothly deformed into noncommutative geometries if certain geometric obstructions vanish. These obstructions can be expressed as a system of partial differential equations relating the metric and the Poisson structure that describes the noncommutativity. I illustrate...

Let L ⊂ J(M) be a Legendrian submanifold of the 1-jet space of a Riemannian n-manifold M . A correspondence is established between rigid flow trees in M determined by L and boundary punctured rigid pseudo-holomorphic disks in T ∗M , with boundary on the projection of L and asymptotic to the double points of this projection at punctures, provided n ≤ 2, or provided n > 2 and the front of L has o...

We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a stringmanifold is a per-Hilbert–Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator.

Let Fm = (M,F ) be a Finsler manifold and G be the Sasaki– Finsler metric on the slit tangent bundle TM0 = TM {0} of M . We express the scalar curvature ρ̃ of the Riemannian manifold (TM0, G) in terms of some geometrical objects of the Finsler manifold Fm. Then, we find necessary and sufficient conditions for ρ̃ to be a positively homogenenous function of degree zero with respect to the fiber coo...

The aim of this paper is to show that holonomy properties of Finsler manifolds can be very different from those of Riemannian manifolds. We prove that the holonomy group of a positive definite non-Riemannian Finsler manifold of non-zero constant curvature with dimension > 2 cannot be a compact Lie group. Hence this holonomy group does not occur as the holonomy group of any Riemannian manifold. ...

This paper investigates the relationship between two fundamental types of objects associated with a connection on a manifold: the existence of parallel semi-Riemannian metrics and the associated holonomy group. Typically in Riemannian geometry, a metric is specified which determines a Levi-Civita connection. Here we consider the connection as more fundamental and allow for the possibility of se...

This paper is motivated by Quantum Field Theory. Whilst the pseudo-Riemannian spacetime manifold enters directly into the formulation of Classical Field Theory, it enters into Quantum Field Theory only through a suitable base for its topology ordered under inclusion. This raises the question as to what extent the topological data of the spacetime manifold are still encoded in the partially orde...

In 1923, Eisenhart 1 gave the condition for the existence of a second-order parallel symmetric tensor in a Riemannian manifold. In 1925, Levy 2 proved that a second-order parallel symmetric nonsingular tensor in a real-space form is always proportional to the Riemannian metric. As an improvement of the result of Levy, Sharma 3 proved that any second-order parallel tensor not necessarily symmetr...