  Families of conformal field theories are naturally endowed with a Riemannian
geometry which is locally encoded by correlation functions of exactly marginal
operators. We show that the curvature of such conformal manifolds can be
computed using Euclidean and Lorentzian inversion formulae, which combine the
operator content of the conformal field theory into an analytic function.
Analogously, operators of fixed dimension define bundles over the conformal
manifold whose curvatures can also be computed using inversion formulae. These
results relate curvatures to integrated four-point correlation functions which
are sensitive only to the behavior of the theory at separated points. We apply
these inversion formulae to derive convergent sum rules expressing the
curvature in terms of the spectrum of local operators and their three-point
function coefficients. We further show that the curvature can smoothly diverge
only if a conserved current appears in the spectrum, or if the theory develops
a continuum. We verify our results explicitly in $2d$ examples. In particular,
for $2d$ (2,2) superconformal field theories we derive a lower bound on the
scalar curvature, which is saturated by free theories when the central charge
is a multiple of three.
