Question about the Laplace-Bertrami operator

Question

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.

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.

Image source: Wikipedia

Answer 1

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 $\hat{g}$.