A general question behind this paper is to explore a good notion for
intrinsic curvature in the framework of noncommutative geometry started by
Alain Connes in the 80s. It has only recently begun (2014) to be comprehended
via the intensive study of modular geometry on the noncommutative two tori. In
this paper, we extend recent results on the modular geometry on noncommutative
two tori to a larger class of noncommutative manifolds: toric noncommutative
manifolds. The first contribution of this work is a pseudo differential
calculus which is suitable for spectral geometry on toric noncommutative
manifolds. As the main application, we derive a general expression for the
modular curvature with respect to a conformal change of metric on toric
noncommutative manifolds. By specializing our results to the noncommutative two
and four tori, we recovered the modular curvature functions found in the
previous works. An important technical aspect of the computation is that it is
free of computer assistance.Comment: 59 pages. The paper was reorganized from the previous versio
