  We discuss a metric description of the divergence of a (projectively) Anosov
flow in dimension 3, in terms of its associated growth rates and give metric
and contact geometric characterizations of when a projectively Anosov flow is
Anosov. Then, we study the symmetries that the existence of an invariant volume
form yields on the geometry of an Anosov flow, from various viewpoints of the
theory of contact hyperbolas, Reeb dynamics and Liouville geometry, and give
characterizations of when an Anosov flow is volume preserving in terms of those
theories. We finally use our study to show that the bi-contact surgery
operations of Salmoiraghi can be applied in an arbitrary small neighborhood of
a periodic orbit of any Anosov flow. In particular, we conclude that the
Goodman surgery of Anosov flows can be performed using a bi-contact surgery of
Salmoiraghi.
