# Why is the Laplacian important in Riemannian geometry?

As I've learned more Riemannian geometry, many of my teachers have said that studying the Laplacian (and its eigenvalues) is very important. But I must admit, I've never fully understood why.

Fundamentally, I would like to know why the Laplacian is important among all differential operators on a Riemannian manifold. I would also like to know what geometric information the Laplacian is supposed to encode.

That being said, I have spent a little time thinking about all this, and my current understanding is as follows:

• ** Somehow, the Laplacian is the "only" isometric invariant "scalar differential operator" on a Riemannian manifold. If true, this statement would completely convince me of its importance. However, I don't know the precise meanings the words in quotes, nor do I have any sense at all of why such a statement would be true.

• An isometric immersion $f \colon S \to M$ is harmonic if and only if it represents a minimal submanifold of $M$. In particular, an isometrically immersed submanifold of $\mathbb{R}^n$ is minimal if and only if its coordinate functions are harmonic.

• The Euler-Lagrange equation for the Dirichlet energy is $\Delta f = 0$. (But why we care about minimzing energy is also somewhat mysterious to me.)

• Weitzenböck formulas comparing two elliptic second-order differential operators (and especially Laplacians) give Bochner-type vanishing theorems.

I should point out that I'm aware that harmonic functions satisfy many of the nice properties that complex-analytic functions do (by virtue of elliptic regularity and maximum principle magic). Still, this doesn't quite tell me why I should care about the Laplace operator itself.

Note: I'm aware of this related question on the eigenvalues of the Laplacian. But again, my interest is in Riemannian geometry; matters of applied mathematics (while interesting) are not my focus right now.

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A fairly complete answer to these questions can be found in (or may be based on) the Introduction to the book of S.Rosenberg "The Laplacian on a Riemannian manifold". –  Yuri Vyatkin Feb 1 '14 at 8:39
OK, I've read the Intro and skimmed Chapter 1. The main point seems to be that the heat flow somehow contains topological and geometric data. As such, it can be used to prove theorems like the Hodge Theorem, Chern-Gauss-Bonnet, and Atiyah-Singer. This has convinced me that the heat flow is a powerful tool in differential geometry, and that the spectrum of $\Delta$ must have some geometric information in it somewhere. But I still feel like I'm missing some things, because I don't feel as though my questions have been answered in full. –  Jesse Madnick Feb 1 '14 at 11:07
Here's something you may appreciate: associated to a differentiable operator is its symbol, which is a tensor that encodes the highest order part of the operator. The symbol of the Laplacian is the metric tensor. –  Eric O. Korman Feb 1 '14 at 16:03
Perhaps someone with a better understanding than me can expand on this, but I believe the Selberg Trace Formula relates the eigenvalues of the Laplacian to the lengths of (primitive, periodic) geodesics in your manifold. If your interest is in Riemannian geometry, I would imagine that should make the Laplacian pretty significant. –  Strawberries Jun 14 '14 at 3:04