  Two years ago, we found the supersymmetric counterpart of the spectral triple
which specified noncommutative geometry. Based on the triple, we derived gauge
vector supermultiplets, Higgs supermultiplets of the minimum supersymmetric
standard model and its action. However, unlike the famous theories of Connes
and his co-workers, the action does not couple to gravity. In this paper, we
obtain the supersymmetric Dirac operator $\mathcal{D}_M^{(SG)}$ on the
Riemann-Cartan curved space replacing derivatives which appear in that of the
triple with the covariant derivatives of general coordinate transformation. We
apply the supersymmetric version of the spectral action principle and
investigate the heat kernel expansion on the square of the Dirac operator. As a
result, we obtain a new supergravity action which does not include the Ricci
curvature tensor.
