---
tag: B7XDAW51Z1
kind: definition
name: Spectral geometry
msc: [58J50, 58J53, 35P15]
defines: QKAFRDPGE4
uses: [7CBT4SM83Y, F4NWZNTRMD, B8TYKV7AA1, 2JK9YMZV2B, TK1PT9FDV0]
sources:
  - url: https://en.wikipedia.org/wiki/Spectral_geometry
---

## Definition

Let $(M, g)$ be a compact [[Riemannian manifold|7CBT4SM83Y]], possibly with boundary and a boundary condition.
Let $\operatorname{spec}(M, g)$ be the [[spectrum of the Laplace–Beltrami operator|B8TYKV7AA1]], the sequence of its eigenvalues with multiplicities.
More generally, one may take the spectrum of another natural elliptic operator, such as the Hodge Laplacian on forms.
*Spectral geometry* is the study of the relations between $\operatorname{spec}(M, g)$ and the geometric invariants of $(M, g)$.
It has two directions:

1. direct problems: estimate or compute the eigenvalues from geometric data;
2. inverse problems: recover geometric data from the spectrum.

## Remarks

Example of a direct result: the [[Lichnerowicz–Obata theorem|TK1PT9FDV0]] bounds the first nonzero eigenvalue below in terms of a positive lower bound on the Ricci curvature.
Example of an inverse result: by [[Weyl's law|2JK9YMZV2B]] the spectrum of a closed manifold determines its dimension and its volume.
The answer to the inverse problem is not complete in general: there are non-isometric manifolds with the same spectrum.

Non-example: the study of the length of a single closed geodesic, without reference to any spectrum, is Riemannian geometry but not spectral geometry.
Geodesic lengths enter spectral geometry only through the length spectrum and trace formulas.