📂Geometry

Geodesic Coordinate Transformation

Definition¹

Let’s define $U \subset \mathbb{R}^{2}$ as an open set. Define $\mathbf{x} : U \to \mathbb{R}^{3}$ as a coordinate chart that satisfies the following:

$$ g_{11} = 1 \quad \text{and} \quad g_{12} = g_{21} = 0 $$

$$ \left[ g_{ij} \right] = \begin{bmatrix} 1 & 0 \\ 0 & g_{22} \end{bmatrix} $$

In this case, $g_{ij}$ is the coefficient of the first fundamental form. Such $\mathbf{x}$ is called a geodesic coordinate chart.
$${}$$ For a surface $M$, if there is a curve $\gamma : [a,b] \to M$ that satisfies $\gamma\left( [a,b] \right) \subset \mathbf{x}\left( U \right)$ and is a $u^{2}-$curve of $\mathbf{x}$, then $\mathbf{x}$ is called the geodesic coordinate chart along $\boldsymbol{\gamma}$.