Conformal structures in noncommutative geometry

Abstract

It is well known that a compact Riemannian spin manifold $(M,g)$ can be reconstructed from its canonical spectral triple $(C^{\infty }(M),L^2(M,\Sigma M),D)$ where $\Sigma M$ denotes the spinor bundle and $D$ the Dirac operator. We show that $g$ can be reconstructed up to conformal equivalence from $(C^{\infty }(M),L^2(M,\Sigma M),\mathrm{sign}(D))$.