First Fundamental Form

The first fundamental form is a way of encoding how distances are measured near each point of a surface. Specifically, it's a type of Riemannian metric that allows you to compute the length of curves on the surface. The first fundamental form is often denoted $I$ or $ds^2$, also known as the line element, and it can be expressed in terms of coordinate differentials $dx$ and $dy$ as follows:

$$I = ds^2 = E dx^2 + 2F dx dy + G dy^2$$

The Infinitesimal Line Element ds = $\sqrt{I}$ Measures Infinitesimal Distances on the Surface

Here, $E$, $F$, and $G$ are functions of the coordinates $x, y$ and describe how to measure distances in the coordinate system defined around each point of the surface. The line element $ds^2$ specifies the infinitesimal distance between two nearby points and serves as a local measure of distance. Furthermore, $ds$, often called the "infinitesimal line element" or "infinitesimal distance," is the square root of $ds^2$ and represents the infinitesimal distance between two points in a mathematically rigorous way.

Metric Tensor

The metric tensor is a generalization of this concept to Riemannian manifolds, which can have any number of dimensions. For a 2D surface embedded in 3D space, the first fundamental form and the metric tensor are effectively the same object. In more general settings, the metric tensor $g$ is a symmetric, positive-definite tensor field that provides a way to compute distances, angles, and areas in a manner that varies smoothly from point to point across the manifold. In local coordinates, the metric tensor $g$ can be represented as a matrix $g_{ij}$, and the line element $ds^2$ of an infinitesimal displacement vector $dx^i$ is given by:

$$ds^2 = g_{ij} dx^i dx^j$$

Here, $g_{ij}$ are the components of the metric tensor, and $dx^i$ are the coordinate differentials. The line element $ds^2$ serves as an infinitesimal "ruler" that measures the distance between infinitesimally close points and is fundamental to the geometry of the manifold. When the manifold is a surface, this expression reduces to the expression for the first fundamental form.

So, in summary, the metric tensor can be thought of as a generalization of the first fundamental form to higher-dimensional spaces and more general coordinate systems. In the specific case of a 2D surface, they are essentially the same mathematical object used to measure distances and angles on the surface. The line element $ds^2$ is a crucial ingredient in this geometry, serving both as a local measure of distance and as a way to generalize these concepts to manifolds of any dimension. The infinitesimal line element $ds$ further refines our understanding, acting as the infinitesimal measure of distance between points.