first fundamental form

first fundamental form

In

classical differential geometry

for embedded two-dimensional

surfaces

M

in

ℝ

3

,
the

Riemannian metric

for

M

induced from the

dot product

of

ℝ

3

is called the

first fundamental form

.

There are various notations for the first fundamental form;
a common notation is

ℐ

,
for the roman letter one. Thus,

ℐ

⁢

(

v

,

w

)

=

v

⋅

w

(1)

for vectors

v

,

w

∈

ℝ

3

. (We consider the

tangent planes

of

M

to be two-dimensional

subspaces

of

ℝ

3

.)

Quadratic form representation

Recall, in

linear algebra

,
that a

symmetric bilinear form

T

over

ℝ

can always be represented by its

quadratic form

Q

:

T

⁢

(

u

,

v

)

=

1

2

⁢

(

Q

⁢

(

u

+

v

)

-

Q

⁢

(

u

)

-

Q

⁢

(

v

)

)

,

Q

⁢

(

w

)

=

T

⁢

(

w

,

w

)

,

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

d

⁢

s

2

=

E

⁢

d

⁢

u

2

+

2

⁢

F

⁢

d

⁢

u

⁢

d

⁢

v

+

G

⁢

d

⁢

v

2

.

(2)

In modern terminology, (

2

) is the quadratic form that
represents the

bilinear form

ℐ

.
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

g

i

⁢

j

, these coefficients
are defined by

E

=

g

11

,

F

=

g

12

=

g

21

,

G

=

g

22

.

The letters

u

and

v

in (

2

) denote

local coordinates

on

M

.
Classically,

d

⁢

u

and

d

⁢

v

meant “infinitesimally small” changes in

u

and

v

,
but in modern

differential geometry

,

d

⁢

u

and

d

⁢

v

have been given a precise
meaning using

differential forms

.

In tensor notation, (

2

) is written as

ℐ

=

E

⁢

d

⁢

u

⊗

d

⁢

u

+

F

⁢

d

⁢

u

⊗

d

⁢

v

+

F

⁢

d

⁢

v

⊗

d

⁢

u

+

G

⁢

d

⁢

v

⊗

d

⁢

v

,

(3)

Although the tensor notation is more clumsy,
it allows us to rigorously justify a change of variables, by the rule

α

*

⁢

(

d

⁢

u

⊗

d

⁢

v

)

=

d

⁢

(

α

*

⁢

u

)

⊗

d

⁢

(

α

*

⁢

v

)

.
See the example below.

The symbol

d

⁢

s

in (

2

) alludes
to

d

⁢

s

2

=

d

⁢

x

2

+

d

⁢

y

2

+

d

⁢

z

2

,

the

infinitesimal

length of a curve. Compare with the modern notation

ℐ

=

d

⁢

x

⊗

d

⁢

x

+

d

⁢

y

⊗

d

⁢

y

+

d

⁢

z

⊗

d

⁢

z

.

(4)

(This is just an alternate way of writing the definition of

ℐ

:
the

restriction

of the dot product on

ℝ

3

.)

Example: sphere

We illustrate an example: we compute the first fundamental form

ℐ

of the sphere

S

2

in

spherical coordinates

(

latitude

/

longitude

system).
We set

x

=

cos

⁡

ϕ

⁢

cos

⁡

θ

y

=

cos

⁡

ϕ

⁢

sin

⁡

θ

z

=

sin

⁡

ϕ

,

and substitute these in (

4

):

ℐ

=

d

⁢

(

cos

⁡

ϕ

⁢

cos

⁡

θ

)

⊗

d

⁢

(

cos

⁡

ϕ

⁢

cos

⁡

θ

)

+

d

⁢

(

cos

⁡

ϕ

⁢

sin

⁡

θ

)

⊗

d

⁢

(

cos

⁡

ϕ

⁢

sin

⁡

θ

)

+

d

⁢

(

sin

⁡

ϕ

)

⊗

d

⁢

(

sin

⁡

ϕ

)

=

(

-

sin

⁡

ϕ

⁢

cos

⁡

θ

⁢

d

⁢

ϕ

-

cos

⁡

ϕ

⁢

sin

⁡

θ

⁢

d

⁢

θ

)

⊗

(

-

sin

⁡

ϕ

⁢

cos

⁡

θ

⁢

d

⁢

ϕ

-

cos

⁡

ϕ

⁢

sin

⁡

θ

⁢

d

⁢

θ

)

+

(

-

sin

⁡

ϕ

⁢

sin

⁡

θ

⁢

d

⁢

ϕ

+

cos

⁡

ϕ

⁢

cos

⁡

θ

⁢

d

⁢

θ

)

⊗

(

-

sin

⁡

ϕ

⁢

sin

⁡

θ

⁢

d

⁢

ϕ

+

cos

⁡

ϕ

⁢

cos

⁡

θ

⁢

d

⁢

θ

)

+

(

cos

⁡

ϕ

⁢

d

⁢

ϕ

)

⊗

(

cos

⁡

ϕ

⁢

d

⁢

ϕ

)

=

(

sin

⁡

ϕ

⁢

cos

⁡

θ

)

2

⁢

d

⁢

ϕ

⊗

d

⁢

ϕ

+

(

cos

⁡

ϕ

⁢

sin

⁡

θ

)

2

⁢

d

⁢

θ

⊗

d

⁢

θ

+

(

sin

⁡

ϕ

⁢

sin

⁡

θ

)

2

⁢

d

⁢

ϕ

⊗

d

⁢

ϕ

+

(

cos

⁡

ϕ

⁢

cos

⁡

θ

)

2

⁢

d

⁢

θ

⊗

d

⁢

θ

+

(

cos

⁡

ϕ

)

2

⁢

d

⁢

ϕ

⊗

d

⁢

ϕ

(note that the cross terms with

d

⁢

ϕ

⊗

d

⁢

θ

and

d

⁢

θ

⊗

d

⁢

ϕ

cancel)

=

sin

2

⁡

ϕ

⁢

d

⁢

ϕ

⊗

d

⁢

ϕ

+

cos

2

⁡

ϕ

⁢

d

⁢

θ

⊗

d

⁢

θ

+

cos

2

⁡

ϕ

⁢

d

⁢

ϕ

⊗

d

⁢

ϕ

=

d

⁢

ϕ

⊗

d

⁢

ϕ

+

cos

2

⁡

ϕ

⁢

d

⁢

θ

⊗

d

⁢

θ

.

Of course this was a very cumbersome calculation; the writing would be simplified
if we had just dropped the

⊗

signs and wrote

d

⁢

ϕ

2

for

d

⁢

ϕ

⊗

d

⁢

ϕ

, etc.
And even then the calculation would be more organized if we computed
the coefficients

g

i

⁢

j

directly.
We only show this kind calculation in order to justify
what exactly
is meant by
the classical expression

d

⁢

ϕ

2

+

cos

2

⁡

ϕ

⁢

d

⁢

θ

2

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

d

⁢

s

2

=

E

⁢

d

⁢

u

2

+

2

⁢

F

⁢

d

⁢

u

⁢

d

⁢

v

+

G

⁢

d

⁢

v

2

then

d

⁢

A

=

E

⁢

G

-

F

2

⁢

d

⁢

u

∧

d

⁢

v

is the area form.
For the sphere, this
is

d

⁢

A

=

cos

2

⁡

ϕ

-

0

⁢

d

⁢

ϕ

∧

d

⁢

θ

=

cos

⁡

ϕ

⁢

d

⁢

ϕ

∧

d

⁢

θ

,

which is just the

formula

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

=

∫

γ

𝑑

s

=

∫

γ

d

⁢

s

2

=

∫

γ

E

⁢

d

⁢

u

2

+

2

⁢

F

⁢

d

⁢

u

⁢

d

⁢

v

+

G

⁢

d

⁢

v

2

,

γ

⁢

(

t

)

=

(

u

,

v

)

,

=

∫

a

b

E

⁢

(

d

⁢

u

d

⁢

t

)

2

+

2

⁢

F

⁢

d

⁢

u

d

⁢

t

⁢

d

⁢

v

d

⁢

t

+

G

⁢

(

d

⁢

v

d

⁢

t

)

2

⁢

𝑑

t

.

Although in practice it is probably easier to directly use

cartesian coordinates

, 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 curves

on a surface.

Example: plane and cylinder

For the plane

ℝ

2

⊂

ℝ

3

with

z

=

0

,
the first fundamental form is just

d

⁢

s

2

=

d

⁢

x

2

+

d

⁢

y

2

.

(5)

For the

cylinder

with the

coordinates

x

=

cos

⁡

u

y

=

sin

⁡

u

z

=

v

the first fundamental form is

d

⁢

s

2

=

d

⁢

u

2

+

d

⁢

v

2

.

(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

isometry

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

:

M

→

N

is an isometry of two surfaces,
and

u

,

v

are coordinates on

M

.
If we use the coordinates

u

′

=

u

∘

Φ

-

1

and

v

′

=

v

∘

Φ

-

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 mapping

:
a diffeomorphism

Φ

:

M

→

N

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 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

Φ

:

M

→

N

is called

equiareal

if

Φ

preserves preserves areas of all subregions of the surfaces.
This amounts to saying that the quantity

E

⁢

G

-

F

2

is

invariant

under

Φ

(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

r

has area

4

⁢

π

⁢

r

2

, because the cylinder that wraps it
also has area

4

⁢

π

⁢

r

2

.

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 curvature

K

⁢

(

p

)

at a point on a surface:

K

=

|

-

1

2

⁢

E

v

⁢

v

+

F

u

⁢

v

-

1

2

⁢

G

u

⁢

u

1

2

⁢

E

u

F

u

-

1

2

⁢

E

v

F

v

-

1

2

⁢

G

u

E

F

1

2

⁢

G

v

F

G

|

-

|

0

1

2

⁢

E

v

1

2

⁢

G

u

1

2

⁢

E

v

E

F

1

2

⁢

G

u

F

G

|

(

E

⁢

G

-

F

2

)

2

where the bars denote the

determinant

, and
the

subscripts

denote

partial derivatives

.

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

embedding

of the surface in

ℝ

3

).

References

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

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