What is spectral invariant? Picture below is from The Spectrum of the Laplacian in Riemannian
Geometry . What is the mean of spectral invariant ? I google  spectral invariant , and find it is a notation of  symplectic geometry . But seemly, there is nothing about  symplectic geometry.





The term spectral invariant refers to an object, such as a function $Z$ of one real variable, defined in terms of a Riemannian manifold $(M, g)$ but that depends only on the spectrum of the $g$-Laplacian on the space of square-integrable functions on $M$.
Isometric manifolds obviously have equal spectral invariants, but the converse is not a priori apparent (and in fact not true, hence the concept of isospectral manifolds and the field of spectral geometry).
