  Recently Dabrowski etc. \cite{DL} obtained the metric and Einstein
functionals by two vector fields and Laplace-type operators over vector
bundles, giving an interesting example of the spinor connection and square of
the Dirac operator. Pf$\ddot{a}$ffle and Stephan \cite{PS1} considered
orthogonal connections with arbitrary torsion on compact Riemannian manifolds
and computed the spectral action. Motivated by the spectral functionals and
Dirac operators with torsion, we give some new spectral functionals which is
the extension of spectral functionals to the noncommutative realm with torsion,
and we relate them to the noncommutative residue for manifolds with boundary.
Our method of producing these spectral functionals is the noncommutative
residue and Dirac operators with torsion.

Differential Geometry