  On a sub-Riemannian manifold, a connection with skew-symmetric torsion is
defined as the unique connection from the class of $N$-connections that has
this property. Two cases are considered separately: sub-Riemannian structure of
even rank, and sub-Riemannian structure of odd rank. The resulting connection,
called the canonical connection, is not a metric connection in the case when
the sub-Riemannian structure is of even rank. The structure of an almost
quasi-Sasakian manifold is defined as an almost contact metric structure of odd
rank that satisfies additional requirements. Namely, it is required that the
canonical connection is a metric connection and that the transversal structure
is a K\"ahler structure. Both the quasi-Sasakian structure and the more general
almost contact metric structure, called an almost quasi-Sasakian structure,
satisfy these requirements. Sufficient conditions are found for an almost
quasi-Sasakian manifold to be an Einstein manifold.
