3
$\begingroup$

I have done a course in Riemannian geometry, where we defined the Laplace-Bertrami operator, but only marginally, so I am now concentrating on examples and exercises. I am familiar with the following formula for the operator.

enter image description here

Question 1: The Wikipedia article says that "In the usual Cartesian coordinates $x^i$ on Euclidean space, the metric is reduced to the Kronecker delta, and one therefore has $|g| = 1$." I dont understand why is the "$g_ij$ missing in this case? And why is the metric reduced to the Kronecker delta?

Question 2: Could you recommend me any sources for examples and calculations of the Laplace-Bertrami operator for some particular spaces?

Thank you for your help.

enter image description here

Image source: Wikipedia

$\endgroup$

1 Answer 1

6
$\begingroup$
  1. The metric $g$ is represented in coordinates by the symmetric positive definite matrix $G = (g_{ij})_{i, j = 1}^{n}$ where $g_{ij}(x) = g(d\phi(x)e_i, d\phi(x)e_j)$, where $\phi$ is your coordinate chart. And common notation is $G^{-1} = (g^{ij})_{i, j = 1}^{n}$ and $g = \det G$. In Euclidean space, using the identity map as coordinates and using the Euclidean metric $g$, we have $g_{ij} = g(e_i, e_j) = e_i \cdot e_j = \delta_{ij}$, i.e. $G = I$. Hence $G^{-1} = I$ and $g = \det I = 1$.

  2. You have the formula, so you can go to all your favorite manifolds to see what the Laplacian looks like. One interesting and useful one is the formula for the Laplacian in polar coordinates on $\mathbb{R}^n$. In this case, with $x = r\omega$, $\omega \in S^{n - 1}$, $$\Delta_{\mathbb{R}^n} = \frac{\partial^2}{\partial r^2} + \frac{n - 1}{r}\frac{\partial}{\partial r} + \frac{1}{r^2}\Delta_{S^{n - 1}}.$$

Edit: Here is some clarification on some of the objects in 1. I use the chart $\phi : O \subset \mathbb{R}^n \to U \subset M$, where $M$ is the manifold. The metric tensor $g \in T^2T^*M$ then has a coordinate representation $\hat{g} = \phi^*g \in T^2T^*O$ defined by $\hat{g}(x)(v, w) = g(\phi(x))(d\phi(x)v, d\phi(x)w)$ for $x \in O$, $v, w \in \mathbb{R}^n$. Now the matrix valued function $G : O \to M(n \times n, \mathbb{R})$ is defined by $g_{ij}(x) = \hat{g}(x)(e_i, e_j) = g(\phi(x))(d\phi(x)e_i, d\phi(x)e_j)$, where $\{e_1, \dots, e_n\}$ is the standard basis of $\mathbb{R}^n$. $G$ is a representation for $\hat{g}$ in the sense that you can go from $\hat{g}$ to $G$ and $G$ to $\hat{g}$ using the formula $\hat{g}(x)(v, w) = \sum_{i, j = 1}^{n}g_{ij}(x)v^iw^j = v \cdot G(x)w$. Using $G$ has several advantages over $\hat{g}$.

$\endgroup$
2
  • $\begingroup$ Thank you! Sorry for being a beginner in this, but could you please clarify: Where do the $e_i, e_j$ live? Are these basis coordinates? And where do the $g_ij$ live? On the manifold or in the tangent space? Similar question about $d \phi(x)$. I am struggling to make this clear. $\endgroup$ Aug 14, 2022 at 15:53
  • 1
    $\begingroup$ @TerezaTizkova I added some clarifications about where everything lives. $\endgroup$
    – Mason
    Aug 14, 2022 at 17:01

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .