  We provide obstructions to the existence of conformally Anosov Reeb flows on
a 3-manifold that partially generalize similar obstructions to Anosov Reeb
flows. In particular, we show $\mathbb{S}^3$ does not admit conformally Anosov
Reeb flows. We also give a Riemannian geometric condition on a metric
compatible with a contact structure implying that a Reeb field is Anosov. From
this we can give curvature conditions on a metric compatible with a contact
structure that implies universal tightness of the contact structure among other
things.
