Normal forms for sub-Lorentzian metrics supported on Engel type distributions

Shortest Paths for Sub-riemannian Metrics on Rank-two Distributions

We study length-minimizing arcs in sub-Riemannian manifolds (M;E;G) whose metric G is de ned on a rank-two bracket-generating distribution E. It is well known that all length-minimizing arcs are extremals, and that these extremals are either \normal" or \abnormal." Normal extremals are locally optimal, in the sense that every su ciently short piece of such an extremal is a minimizer. The questi...

Variational Equations on Mixed Riemannian-lorentzian Metrics

A class of elliptic-hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, which is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fiel...

On Lorentzian causality with continuous metrics

We present a systematic study of causality theory on Lorentzian manifolds with continuous metrics. Examples are given which show that some standard facts in smooth Lorentzian geometry, such as light-cones being hypersurfaces, are wrong when metrics which are merely continuous are considered. We show that existence of time functions remains true on domains of dependence with continuous metrics, ...

On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1

The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using Pontryagin Maximum Principle, we treat Riemannian and sub-Riemannian cases in an unified way and obtain some algebraic necessary conditions for the geodesic equivalence of (sub-)Riemannian metrics. In this way first we obtain a new...

I the Distributions of Quadratic Forms on Normal Random Variables