This paper proposes a puzzling but somewhat vaguely formulated 
criterion for deducing isometry of compact Riemannian manifolds 
from discrete sets of spectral data (that is, the set of eigenvalues, 
and a sequence constructed from eigenfunctions). Unfortunately, 
the paper is written in a way that makes it hard to check the claim 
(for instance, what is the meaning of ``orthonormal basis map 𝐹 that 
preserves {𝑀^{𝑖,𝑗,𝑘}}" -- that map is supposed to take smooth functions 
to smooth functions -- I assume it is a map between the basis which 
then becomes a map on functions using the Fourier expansions? But 
then at some other place (at the beginning of Sect.3) F is map from 
one manifold to another. This vagueness discourages careful reading.

In addition, it is not clear what is the merit of this result in terms of proving 
some new facts in spectral geometry. Having some applications and a 
serious rewrite are unfortunately necessary before the paper can be 
seriously considered.