Heat kernel on a Riemannian manifold

Summary

The heat kernel of a Riemannian manifold is the fundamental solution of the heat equation $\partial_t u + \Delta u = 0$, where $\Delta$ is the Laplace–Beltrami operator. On a closed manifold it is the integral kernel of $e^{-t\Delta}$ and is the sum of $e^{-t\lambda_j} \phi_j(x) \phi_j(y)$ over an orthonormal basis of eigenfunctions. Its behaviour for small time encodes local geometry, such as volume and curvature, and its trace encodes the spectrum. It is the main tool of the heat equation approach to index theory and spectral geometry.