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.
2026-03-28 02:03:28.1774663408
What is spectral invariant?
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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).