first fundamental form
There are various notations for the first fundamental form; a common notation is , for the roman letter one. Thus,
Quadratic form representation
for any vectors and . This process may be applied to the first fundamental form, and classically, the first fundamental form is expressed as
In modern terminology, (2) is the quadratic form that represents the bilinear form . The use of the letters for the coefficients of the quadratic form is traditional, and dates back to Gauss; in terms of the metric tensor , these coefficients are defined by , , .
The letters and in (2) denote local coordinates on . Classically, and meant “infinitesimally small” changes in and , but in modern differential geometry, and have been given a precise meaning using differential forms.
In tensor notation, (2) is written as
Although the tensor notation is more clumsy, it allows us to rigorously justify a change of variables, by the rule . See the example below.
and substitute these in (4):
|(note that the cross terms with and cancel)|
Of course this was a very cumbersome calculation; the writing would be simplified if we had just dropped the signs and wrote for , etc. And even then the calculation would be more organized if we computed the coefficients directly. We only show this kind calculation in order to justify what exactly is meant by the classical expression
for the first fundamental form of the sphere.
Use of first fundamental form to compute lengths and areas
The first fundamental form is related to the area form. If
is the area form. For the sphere, this is
The first fundamental form itself may be used to find the length of a curve on a surface , when is parameterized by local coordinates:
Although in practice it is probably easier to directly use cartesian coordinates, rather than the above expressions, to compute the length of , the first fundamental form plays an essential role in the theoretical investigation of the lengths of curves on a surface.
Example: plane and cylinder
For the plane with , the first fundamental form is just
Relation with isometric maps
Notice that looks the same as (5) after renaming the variables. This is evidence that the plane and cylinder should be locally isometric: a flat sheet can be rolled into a cylinder. An isometry between two surfaces, by definition, preserves the metric on the two surfaces, so an isometry preserves the first fundamental form.
Suppose is an isometry of two surfaces, and are coordinates on . If we use the coordinates and on , then the first fundamental form of is obtained by taking the first fundamental form of and renaming to .
Relation with conformal, equiareal maps
There is also a notion of a conformal mapping: a diffeomorphism is called conformal if preserves the first fundamental form up to a non-zero constant of proportionality. (The proportion may vary at each point of and .) It may be verified that this is the same as saying that preserves the angles of intersecting tangent vectors.
For example, the stereographic projection from the sphere to the plane is conformal.
Yet another notion is that of an equiareal mapping: a diffeomorphism is called equiareal if preserves preserves areas of all subregions of the surfaces. This amounts to saying that the quantity is invariant under (provided we rename the variables as explained above).
For example, the projection of the sphere to the cylinder wrapping it is equiareal. This fact was used by Archimedes to show the sphere of radius has area , because the cylinder that wraps it also has area .
If is both conformal and equiareal, then it is an isometry. As a well-known example, a sphere is not isometric to the plane, not even locally, so we cannot draw maps of the Earth that preserve both directions and relative proportion of lands. We must give up at least one of these properties: e.g. the Mercator projection preserves direction only; maps with Mercator look “strange” the first time one sees them, because such maps do not preserve area.
Relation with Gaussian curvature
There is a formula for the Gaussian curvature at a point on a surface:
This formula is known as Brioschi’s formula; Brioschi had stated it without proof in 1854, and later it was calculated by Gauss.
The immediate corollary of this strange formula is:
Theorem 1 (Theorema Egregium).
The Gaussian curvature of a surface is unchanged under isometries (because it only depends on the first fundamental form).
|Title||first fundamental form|
|Date of creation||2013-03-22 15:28:38|
|Last modified on||2013-03-22 15:28:38|
|Last modified by||stevecheng (10074)|