  We study how the Riemannian structure on a manifold can be usefully
reconstructed from its codifferential $\delta$, including a formula
$\nabla_\omega\eta={1\over 2}(
\delta(\omega\eta)-(\delta\omega)\eta+\omega(\delta\eta) +L_\omega(\eta)+i_\eta
d \omega)$ for the Levi-Civita covariant derivative in terms of 1-forms, where
$L, i$ are respectively the Lie derivative and interior product along the
corresponding vector fields. The covariant derivative extends naturally along
forms of any degree and to possibly degenerate $(\ ,\ )$. In the nondegenerate
case, $\delta$ makes the exterior algebra into a BV algebra. In the invertible
case we show that ${\rm Ricci}=-{1\over 2}\Delta g$ where the Hodge Laplacian
$\Delta$ extends in a natural way to act on the metric.
  Our results come from a new way of thinking about metrics and connections as
a kind of cocycle data for central extensions of differential graded algebras
(DGAs), a theory which we introduce. We show that any cleft extension of the
exterior algebra $\Omega(M)$ on a manifold is associated to a
possibly-degenerate metric and covariant derivative. Those for which $d$ is not
deformed up to isomorphism correspond to the Levi-Civita case. We provide a
construction for such extensions both of classical DGAs and of already
non-graded-commutative DGAs, thereby constructing a class of bimodule covariant
derivatives via a kind of quantum analogue of the Koszul formula.
  We also provide a semidirect product of any differential graded algebra by
the quantum differential algebra $\Omega(t,d t)$ in one variable, to introduce
a noncommutative `time'. Composing these two constructions recovers a previous
differential quantisation of $M\times R$.
