Page 304 of Noncommutative Geometry and the Standard Model of Elementary Particle Physics

We cannot (completely) hear the shape of a drum, even if it is spinorial. Two metric fields with the same collection of eigenvalues are called isospectral. There exist Dirac isospectral deformations; continuous 1-parameter families of mutually non- isometric metrics with the same Dirac spectrum have been constructed. They are of the form $M_s = G/F_s$, $s \in \mathbb{C}$, with $G$ a nilpotent group (e.g. the Heisenberg group) and $F_s$ a nilpotent subgroup. Also, there exist known examples of Laplace-isospectral 4-dimensional flat tori which are also Dirac-isospectral (at least for the trivial spin structure).

No, we cannot (completely) hear the shape of a drum, even if it is spinorial. Two metric fields with the same collection of eigenvalues are called isospectral. There exist Dirac isospectral deformations; continuous 1-parameter families of mutually non- isometric metrics with the same Dirac spectrum have been constructed. They are of the form $M_s = G/F_s$, $s \in \mathbb{C}$, with $G$ a nilpotent group (e.g. the Heisenberg group) and $F_s$ a nilpotent subgroup. Also, there exist known examples of Laplace-isospectral 4-dimensional flat tori which are also Dirac-isospectral (at least for the trivial spin structure).

Noncommutative Geometry and the Standard Model of Elementary Particle Physics (page 304).