  Every volume-preserving centre-bunched fibred partially hyperbolic system
with 2-dimensional centre either (1) has two distinct centre Lyapunov
exponents, or (2) exhibits an invariant continuous line field (or pair of line
fields) tangent to the centre leaves, or (3) admits a continuous conformal
structure on the centre leaves invariant under both the dynamics and the stable
and unstable holonomies. The last two alternatives carry strong restrictions on
the topology of the centre leaves: (2) can only occur on tori, and for (3) the
centre leaves must be either tori or spheres. Moreover, under some additional
conditions, such maps are rigid, in the sense that they are topologically
conjugate to specific algebraic models. When the system is symplectic (1)
implies that the centre Lyapunov exponents are non-zero, and thus the system is
(non-uniformly) hyperbolic.
