General questions on the eigenfunctions of Laplacian and Dirac operators We know that the eigenvalues of the Laplacian contains a lot of information of a Riemannian manifold, but they do not determine the full information ( Hearing the shape of a drum). And the eigenfunctions of the Laplacian seem to have much more information (see the reference). Now my question is that whether the eigenfunctions of the Dirac operator would contain more information than that of Laplacian, since it seems to me that the Dirac operator is a more refined version of the Laplacian ( a Riemmanian manifold could have several spin structure).
Here the Laplacian means the Laplace-Beltrami operator and the Dirac operator means the Dirac operator on the spinor bundle. Thank you.
 A: 
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).
A: Most of the questions raised above are answered in the article "The Dirac Operator on Nilmanifolds and Collapsing Circle Bundles" by Christian Bär and myself published in Annals of Global Analysis and Geometry
June 1998, Volume 16, Issue 3, pp 221–253. http://link.springer.com/article/10.1023/A:1006553302362.
A preprint version is available on the arxive as https://arxiv.org/abs/math/9801091.
All examples of Laplace-isospectral tori are also Dirac-isospectral 
for the trivial spin structure.
In the reference above we constructed Dirac-isospectral 
families of 3-step nilmanfolds, inspired by a construction by Ruth Gornet.
There are also families which are Dirac-isospectral for some spin structures 
and not Dirac-isospectral for other spin structures.
