  We consider a transversally conformal foliation $\mathcal{F}$ of a closed
manifold $M$ endowed with a smooth Riemannian metric whose restriction to each
leaf is negatively curved. We prove that it satisfies the following dichotomy.
Either there is a transverse holonomy-invariant measure for $\mathcal{F}$, or
the foliated geodesic flow admits a finite number of physical measures, which
have negative transverse Lyapunov exponents and whose basin cover a set full
for the Lebesgue measure. We also give necessary and sufficient conditions for
the foliated geodesic flow to be partially hyperbolic in the case where the
foliation is transverse to a projective circle bundle over a closed hyperbolic
surface.
