No tensorial spectral Einstein functional exists for nonzero antisymmetric torsion, so spectral geometry forces torsion to vanish.
Kac, Can One Hear the Shape of a Drum?, The American Mathematical Monthly 73, 1 (1966)
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An impediment to torsion from spectral geometry
No tensorial spectral Einstein functional exists for nonzero antisymmetric torsion, so spectral geometry forces torsion to vanish.