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We examine the metric and Einstein bilinear functionals of differential formsintroduced in Adv.Math.,Vol.427,(2023)1091286, for Hodge-Dirac operator$d+\delta$ on an oriented even-dimensional Riemannian manifold. We show thatthey reproduce these functionals for the canonical Dirac operator on a spinmanifold up to a numerical factor. Furthermore, we demonstrate that theassociated spectral triple is spectrally closed, which implies that it istorsion-free.

Spectral Metric and Einstein Functionals for Hodge-Dirac operator