first fundamental form

In classical differential geometry for embedded two-dimensional surfaces M in 3, the Riemannian metricMathworldPlanetmath for M induced from the dot productMathworldPlanetmath of 3 is called the first fundamental formMathworldPlanetmath.

There are various notations for the first fundamental form; a common notation is , for the roman letter one. Thus,

(v,w)=vw (1)

for vectors v,w3. (We consider the tangent planesMathworldPlanetmath of M to be two-dimensional subspacesPlanetmathPlanetmathPlanetmath of 3.)

Quadratic form representation

Recall, in linear algebra, that a symmetric bilinear formMathworldPlanetmath T over can always be represented by its quadratic formMathworldPlanetmath Q:


for any vectors u and v. This process may be applied to the first fundamental form, and classically, the first fundamental form is expressed as

ds2=Edu2+2Fdudv+Gdv2. (2)

In modern terminology, (2) is the quadratic form that represents the bilinear formMathworldPlanetmathPlanetmath . The use of the letters E,F,G for the coefficients of the quadratic form is traditional, and dates back to Gauss; in terms of the metric tensor gij, these coefficients are defined by E=g11, F=g12=g21, G=g22.

The letters u and v in (2) denote local coordinates on M. Classically, du and dv meant “infinitesimally small” changes in u and v, but in modern differential geometryMathworldPlanetmath, du and dv have been given a precise meaning using differential formsMathworldPlanetmath.

In tensor notation, (2) is written as

=Edudu+Fdudv+Fdvdu+Gdvdv, (3)

Although the tensor notation is more clumsy, it allows us to rigorously justify a change of variables, by the rule α*(dudv)=d(α*u)d(α*v). See the example below.

The symbol ds in (2) alludes to


the infinitesimalMathworldPlanetmathPlanetmath length of a curve. Compare with the modern notation

=dxdx+dydy+dzdz. (4)

(This is just an alternate way of writing the definition of : the restrictionPlanetmathPlanetmath of the dot product on 3.)

Example: sphere

We illustrate an example: we compute the first fundamental form of the sphere S2 in spherical coordinatesMathworldPlanetmath (latitude/longitude system). We set

x =cosϕcosθ
y =cosϕsinθ
z =sinϕ,

and substitute these in (4):

(note that the cross terms with dϕdθ and dθdϕ cancel)

Of course this was a very cumbersome calculation; the writing would be simplified if we had just dropped the signs and wrote dϕ2 for dϕdϕ, etc. And even then the calculation would be more organized if we computed the coefficients gij 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


which is just the formulaMathworldPlanetmathPlanetmath given in calculus for evaluating surface integrals on the sphere using spherical coordinates.

The first fundamental form itself may be used to find the length s of a curve γ on a surface M, when γ is parameterized by local coordinates:

s =γ𝑑s=γds2

Although in practice it is probably easier to directly use cartesian coordinatesMathworldPlanetmath, rather than the above expressions, to compute the length of γ, the first fundamental form plays an essential role in the theoretical investigation of the lengths of curvesPlanetmathPlanetmath on a surface.

Example: plane and cylinder

For the plane 23 with z=0, the first fundamental form is just

ds2=dx2+dy2. (5)

For the cylinderMathworldPlanetmath with the coordinatesMathworldPlanetmathPlanetmath

x =cosu
y =sinu
z =v

the first fundamental form is

ds2=du2+dv2. (6)

Relation with isometric maps

Notice that (6) 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 isometryMathworldPlanetmath between two surfaces, by definition, preserves the metric on the two surfaces, so an isometry preserves the first fundamental form.

Of course, (5) and (6) are expressions of the first fundamental form in local coordinates of two different surfaces, so it makes no sense to say they are equal. But it is not hard to see that:

Suppose Φ:MN is an isometry of two surfaces, and u,v are coordinates on M. If we use the coordinates u=uΦ-1 and v=vΦ-1 on N, then the first fundamental form of N is obtained by taking the first fundamental form of M and renaming u,v to u,v.

Relation with conformal, equiareal maps

There is also a notion of a conformal mappingPlanetmathPlanetmath: a diffeomorphism Φ:MN is called conformal if Φ preserves the first fundamental form up to a non-zero constant of proportionality. (The proportion may vary at each point of M and N.) It may be verified that this is the same as saying that Φ preserves the angles of intersecting tangent vectorsMathworldPlanetmath.

For example, the stereographic projection from the sphere to the plane is conformal.

Yet another notion is that of an equiareal mapping: a diffeomorphism Φ:MN is called equiareal if Φ preserves preserves areas of all subregions of the surfaces. This amounts to saying that the quantity EG-F2 is invariantMathworldPlanetmath under Φ (provided we rename the variables as explained above).

For example, the projectionMathworldPlanetmath of the sphere to the cylinder wrapping it is equiareal. This fact was used by Archimedes to show the sphere of radius r has area 4πr2, because the cylinder that wraps it also has area 4πr2.

If Φ 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 curvatureMathworldPlanetmath K(p) at a point on a surface:


where the bars denote the determinantMathworldPlanetmath, and the subscripts denote partial derivativesMathworldPlanetmath.

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).

This theorem is not obvious, since the usual definitions of the Gaussian curvature are not invariant (they depend on the particular embeddingPlanetmathPlanetmath of the surface in 3).


  • 1 Michael Spivak. A Comprehensive Introduction to Differential Geometry, volumes I and II. Publish or Perish, 1979.
  • 2 Andrew Pressley. Elementary Differential Geometry. Springer-Verlag, 2003.
Title first fundamental form
Canonical name FirstFundamentalForm
Date of creation 2013-03-22 15:28:38
Last modified on 2013-03-22 15:28:38
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 7
Author stevecheng (10074)
Entry type Definition
Classification msc 53B21
Classification msc 53B20
Related topic SecondFundamentalForm
Related topic TiltCurve