  We introduce a generalization of Goodman surgery to the category of
projectively Anosov flows. This construction is performed along a knot that is
simultaneously Legendrian and transverse for a supporting bi-contact structure.
If the flow is Anosov there is a particular class of supporting bi-contact
structures that induce Lorentzian metrics satisfying Barbot's criterion of
hyperbolicity. Foulon and Hasselblatt construct new contact Anosov flows by
surgery from a geodesic flow. We generalize their result showing that in any
contact Anosov flow there is a family of Legendrian knots that can be used to
produce new contact Anosov flows by surgery. Outside of the realm of Anosov
flows we generate new examples of projectively Anosov flows on hyperbolic
3-manifolds. These flows contain an invariant submanifold of genus g>0. We also
give some application to contact geometry: we interpret the bi-contact surgery
in terms of classic contact-Legendrian surgery and admissible-inadmissible
transverse surgery and we deduce some (hyper)tightness result for contact and
transverse surgeries.
