b-Completion of pseudo-Hermitian manifolds

We study the interrelation among pseudohermitian and Lorentzian geometry as prompted by the existence of the Fefferman metric. Specifically for any nondegenerate CR manifold $M$ we build its $b$-boundary $\dot{M}$. This arises as a $S^1$ quotient of the $b$-boundary of the (total space of the canonical circle bundle over $M$ endowed with the) Fefferman metric. Points of $\dot{M}$ are shown to be endpoints of $b$-incomplete curves. A class of inextensible integral curves of the Reeb vector on a pseudo-Einstein manifold is shown to have an endpoint on the $b$-boundary provided that the horizontal gradient of the pseudohermitian scalar curvature satisfies an appropriate boundedness condition.