# First fundamental form

metric form, of a surface

A quadratic form in the differentials of the coordinates on the surface that determines the intrinsic geometry of the surface in a neighbourhood of a given point.

Let the surface be defined by the equation

where and are coordinates on the surface, while

is the differential of the position vector along the chosen direction of displacement from a point to an infinitesimally close point (see Fig. a).

Figure: f040450a

The square of the principal linear part of the increment of the length of the arc can be expressed in terms of the square of the differential :

and is called the first fundamental form of the surface. The coefficients in the first fundamental form are usually denoted by

or, in tensor symbols,

The tensor is called the first fundamental, or metric, tensor of the surface. The first fundamental form is a positive-definite form at regular points on the surface:

The first fundamental form characterizes the metric properties of the surface: Knowledge of the first fundamental form enables one to calculate arc lengths on the surface:

where is a parameter on the curve; angles between curves on the surface:

where and are the directions of the tangent vectors to the curves (see Fig. b); and areas of regions on the surface:

Figure: f040450b

The form of the coefficients of the first fundamental form substantially depends on the choice of the coordinates. The first fundamental form has so-called orthogonal form

in orthogonal coordinates, canonical form

in semi-geodesic coordinates, and isothermal (isometric) form in isothermal coordinates:

Sometimes a surface is characterized by special forms of the first fundamental form. For example, a Liouville surface is characterized by:

The first fundamental form is a bending invariant for the surface: The Gaussian curvature at a given point can be calculated from the coefficients of the first fundamental form and their derivatives only (Gauss' theorem).

See Fundamental forms of a surface for the relation between the first fundamental form and other quadratic forms, as well as for literature.