  We first propose a conformal geometry for Connes-Landi noncommutative
manifolds and study the associated scalar curvature. The new scalar curvature
contains its Riemannian counterpart as the commutative limit. Similar to the
results on noncommutative two tori, the quantum part of the curvature consists
of actions of the modular derivation through two local curvature functions.
Explicit expressions for those functions are obtained for all even dimensions
(greater than two). In dimension four, the one variable function shows striking
similarity to the analytic functions of the characteristic classes appeared in
the Atiyah-Singer local index formula, namely, it is roughly a product of the
$j$-function (which defines the $\hat A$-class of a manifold) and an
exponential function (which defines the Chern character of a bundle). By
performing two different computations for the variation of the Einstein-Hilbert
action, we obtain a deep internal relations between two local curvature
functions. Straightforward verification for those relations gives a strong
conceptual confirmation for the whole computational machinery we have developed
so far, especially the Mathematica code hidden behind the paper.
