  In a previous paper we established Cwikel-type estimates on noncommutative
tori and used them to get analogues in this setting of the
Cwikel-Lieb-Rozenblum (CLR) and Lieb-Thirring inequalities for negative
eigenvalues of fractional Schr\"odinger operators. In this paper, we focus on
"curved" NC tori, where the role of the usual Laplacian is played by
Laplace-Beltrami operators associated with arbitrary Riemannian metrics. The
Cwikel-type estimates of our previous paper are extended to pseudodifferential
operators and powers of Laplace-Beltrami operators. There are several
applications of these estimates. First, we get $L_p$-versions of the usual
formula for the trace of \psidos\ on NC tori, i.e., for combinations of
\psidos\ with $L_p$-position operators. Next, we get $L_p$-versions of the
analogues for NC tori Connes' trace theorem and Connes' integration formula.
They give formulas for the NC integrals (a.k.a.\ Dixmier traces) of products of
$L_p$-position operators with \psidos\ or powers of the Laplace-Beltrami
operators. Moreover, by combining our Cwikel-type estimates with suitable
versions of the Birman-Schwinger principle we get versions of the CLR and
Lieb-Thirring inequalities for negative eigenvalues of fractional Schr\"odinger
operators associated with powers of Laplace-Beltrami operators and
$L_p$-potentials. As in the original Euclidean case the Lieb-Thirring
inequalities imply a dual Sobolev inequality for orthonormal families. Finally,
we discuss spectral asymptotics and semiclassical Weyl's laws for the our
classes of operators on curved NC tori. This superseded a previous conjecture
in our previous paper.
