second fundamental form

second fundamental form

In

classical differential geometry

the

second fundamental form

is a

symmetric bilinear form

defined on a

differentiable

surface

M

embedded in

ℝ

3

, which in some sense
measures the

curvature

of

M

in space.

To construct the second fundamental form requires a small digression.
After the digression we will discuss how it relates to the curvature of

M

.

Construction of the second fundamental form

Consider the

tangent planes

T

p

⁢

M

of the surface

M

for each point

p

∈

M

.
There are two

unit normals

to

T

p

⁢

M

.
Assuming

M

is orientable, we can choose one of these unit normals (

http://planetmath.org/MutualPositionsOfVectors

),

n

⁢

(

p

)

,
so that

n

⁢

(

p

)

varies smoothly with

p

.

Since

n

⁢

(

p

)

is a

unit vector

in

ℝ

3

, it may be considered
as a point on the sphere

S

2

⊂

ℝ

3

. Then we
have a map

n

:

M

→

S

2

. It is called the

normal map

or

Gauss map

.

The

second fundamental form

is the

tensor field

ℐ

⁢

ℐ

on

M

defined
by

ℐ

⁢

ℐ

p

⁢

(

ξ

,

η

)

=

-

⟨

D

⁡

n

p

⁢

(

ξ

)

,

η

⟩

,

ξ

,

η

∈

T

p

⁢

M

,

(1)

where

⟨

,

⟩

is the

dot product

of

ℝ

3

,
and we consider the tangent planes of surfaces in

ℝ

3

to be

subspaces

of

ℝ

3

.

The

linear transformation

D

⁡

n

p

is in reality the

tangent

mapping

D

⁡

n

p

:

T

p

⁢

M

→

T

n

⁢

(

p

)

⁢

S

2

,
but since

T

n

⁢

(

p

)

⁢

S

2

=

T

p

⁢

M

by the definition of

n

,
we prefer to think of

D

⁡

n

p

as

D

⁡

n

p

:

T

p

⁢

M

→

T

p

⁢

M

.

The

tangent map

D

⁡

n

, is often called the

Weingarten map

.

Proposition 1

.

The second fundamental form is a symmetric form.

Proof.

This is a

computation

using a

coordinate chart

σ

for

M

.
Let

u

,

v

be the corresponding names for the

coordinates

.
From the equation

⟨

n

,

∂

⁡

σ

∂

⁡

v

⟩

=

0

,

differentiating with respect to

u

using the

product rule

gives

⟨

n

,

∂

2

⁡

σ

∂

⁡

u

⁢

∂

⁡

v

⟩

=

-

⟨

∂

⁡

n

∂

⁡

u

,

∂

⁡

σ

∂

⁡

v

⟩

=

-

⟨

D

⁡

n

⁢

(

∂

⁡

σ

∂

⁡

u

)

,

∂

⁡

σ

∂

⁡

v

⟩

=

ℐ

⁢

ℐ

⁢

(

∂

⁡

σ

∂

⁡

u

,

∂

⁡

σ

∂

⁡

v

)

.

(2)

(The second equality follows from the definition of the tangent map

D

⁡

n

.)
Reversing the roles of

u

,

v

and repeating the last

derivation

,
we obtain also:

⟨

n

,

∂

2

⁡

σ

∂

⁡

u

⁢

∂

⁡

v

⟩

=

⟨

n

,

∂

2

⁡

σ

∂

⁡

v

⁢

∂

⁡

u

⟩

=

ℐ

⁢

ℐ

⁢

(

∂

⁡

σ

∂

⁡

v

,

∂

⁡

σ

∂

⁡

u

)

.

(3)

Since

∂

⁡

σ

/

∂

⁡

u

and

∂

⁡

σ

/

∂

⁡

v

form a basis for

T

p

⁢

M

, combining
(

2

) and (

3

) proves that

ℐ

⁢

ℐ

is

symmetric

.
∎

In view of

Proposition

1

,
it is customary to regard the second fundamental form
as a

quadratic form

,
as it done with the

first fundamental form

.
Thus, the second fundamental form is referred to with the following expression

1

1

Unfortunately the

coefficient

M

here clashes with our use of the letter

M

for
the surface (manifold), but whenever we write

M

, the

context

should make clear
which meaning is intended.
The use of the symbols

L

,

M

,

N

for the coefficients of the second fundamental form
is standard, but probably was established long before
anyone thought about manifolds.

:

L

⁢

d

⁢

u

2

+

2

⁢

M

⁢

d

⁢

u

⁢

d

⁢

v

+

N

⁢

d

⁢

v

2

.

Compare with the tensor notation

ℐ

⁢

ℐ

=

L

⁢

d

⁢

u

⊗

d

⁢

u

+

M

⁢

d

⁢

u

⊗

d

⁢

v

+

M

⁢

d

⁢

v

⊗

d

⁢

u

+

N

⁢

d

⁢

v

⊗

d

⁢

v

.

Or in matrix form (with respect to the coordinates

u

,

v

),

ℐ

⁢

ℐ

=

(

L

M

M

N

)

.

Curvature of curves on a surface

Let

γ

be a curve

lying on

the surface

M

, parameterized by arc-length.
Recall that the curvature

κ

⁢

(

s

)

of

γ

at

s

is

γ

′′

⁢

(

s

)

. If we want to measure the curvature of the surface, it is natural
to consider the component of

γ

′′

⁢

(

s

)

in the normal

n

⁢

(

γ

⁢

(

s

)

)

.
Precisely, this quantity is

⟨

γ

′′

⁢

(

s

)

,

n

⁢

(

γ

⁢

(

s

)

)

⟩

,

and is called the

normal curvature

of

γ

on

M

.

So to study the curvature of

M

, we ignore the component of the curvature of

γ

in the tangent plane of

M

.
Also, physically speaking, the normal curvature is proportional to the

acceleration

required to keep a moving particle

on

the surface

M

.

We now come to the motivation for defining the second fundamental form:

Proposition 2

.

Let

γ

be a curve on

M

, parameterized by arc-length,
and

γ

⁢

(

s

)

=

p

.
Then

⟨

γ

′′

⁢

(

s

)

,

n

⁢

(

p

)

⟩

=

ℐ

⁢

ℐ

⁢

(

γ

′

⁢

(

s

)

,

γ

′

⁢

(

s

)

)

.

Proof.

From the equation

⟨

n

⁢

(

γ

⁢

(

s

)

)

,

γ

′

⁢

(

s

)

⟩

=

0

,

differentiate

with respect to

s

:

⟨

n

⁢

(

γ

⁢

(

s

)

)

,

γ

′′

⁢

(

s

)

⟩

=

-

⟨

d

d

⁢

s

⁢

n

⁢

(

γ

⁢

(

s

)

)

,

γ

′

⁢

(

s

)

⟩

=

-

⟨

D

⁡

n

⁢

(

γ

′

⁢

(

s

)

)

,

γ

′

⁢

(

s

)

⟩

=

ℐ

⁢

ℐ

⁢

(

γ

′

⁢

(

s

)

,

γ

′

⁢

(

s

)

)

.

∎

It is now time to mention an important

consequence

of Proposition

1

:
the fact that

ℐ

⁢

ℐ

is symmetric means that

-

D

⁡

n

is

self-adjoint

with respect to the

inner product

ℐ

(the first fundamental form). So, if

-

D

⁡

n

is expressed
as a matrix with orthonormal coordinates (with respect to

ℐ

),
then the matrix is symmetric. (The minus sign in front of

D

⁡

n

is to make the

formulas

work out nicely.)

Certain

theorems

in

linear algebra

tell us
that,

-

D

⁡

n

p

being self-adjoint, it has an

orthonormal basis

of

eigenvectors

e

1

,

e

2

with corresponding

eigenvalues

κ

1

≤

κ

2

.
These eigenvalues are called the

principal curvatures

of

M

at

p

.
The eigenvectors

e

1

,

e

2

are the

principal directions

.
The terminology is justified by the following theorem:

Theorem 1

(Euler’s Theorem).

The normal curvature of a curve

γ

has the form

⟨

γ

′′

⁢

(

s

)

,

n

⁢

(

p

)

⟩

=

κ

1

⁢

cos

2

⁡

θ

+

κ

2

⁢

sin

2

⁡

θ

,

p

=

γ

⁢

(

s

)

.

It follows that the

minimum

possible normal curvature is

κ

1

,
and the

maximum

possible is

κ

2

.

Proof.

Since

e

1

,

e

2

form an orthonormal basis for

T

p

⁢

M

, we may write

γ

′

⁢

(

s

)

=

cos

⁡

θ

⁢

e

1

+

sin

⁡

θ

⁢

e

2

for some angle

θ

.
Then

⟨

γ

′′

⁢

(

s

)

,

n

⁢

(

p

)

⟩

=

ℐ

⁢

ℐ

⁢

(

γ

′

⁢

(

s

)

,

γ

′

⁢

(

s

)

)

=

⟨

-

D

⁡

n

p

⁢

(

γ

′

⁢

(

s

)

)

,

γ

′

⁢

(

s

)

⟩

=

⟨

κ

1

⁢

cos

⁡

θ

⁢

e

1

+

κ

2

⁢

sin

⁡

θ

⁢

e

2

,

cos

⁡

θ

⁢

e

1

+

sin

⁡

θ

⁢

e

2

⟩

=

κ

1

⁢

cos

2

⁡

θ

+

κ

2

⁢

sin

2

⁡

θ

.

∎

Matrix representations of second fundamental form and Weingarten map

At this point, we should find the explicit prescriptions
for calculating the second fundamental form and the Weingarten map.

Let

σ

be a coordinate chart for

M

,
and

u

,

v

be the names of the coordinates.
For a test vector

ξ

∈

T

p

⁢

M

, we write

ξ

u

and

ξ

v

for the

u

,

v

coordinates of

ξ

.

We compute the matrix

W

for

-

D

⁡

n

in

u

,

v

-coordinates.
We have

(

ξ

u

ξ

v

)

⁢

(

L

M

M

N

)

⁢

(

ξ

u

ξ

v

)

=

ℐ

⁢

ℐ

⁢

(

ξ

,

ξ

)

=

⟨

-

D

⁡

n

⁢

(

ξ

)

,

ξ

⟩

=

(

Q

⁢

(

ξ

u

ξ

v

)

)

T

⁢

Q

⁢

W

⁢

(

ξ

u

ξ

v

)

=

(

ξ

u

ξ

v

)

⁢

(

Q

T

⁢

Q

)

⁢

W

⁢

(

ξ

u

ξ

v

)

,

where

Q

is the matrix that changes from

u

,

v

-coordinates to orthonormal
coordinates for

T

p

⁢

M

— this is necessary to compute the inner product.
But

Q

T

⁢

Q

=

(

E

F

F

G

)

=

ℐ

(the first fundamental form),

because

Q

is the matrix
with columns

∂

⁡

σ

/

∂

⁡

u

and

∂

⁡

σ

/

∂

⁡

v

expressed in orthonormal coordinates.

(More to be written…)

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

second fundamental form

Canonical name

SecondFundamentalForm

Date of creation

2013-03-22 15:29:02

Last modified on

2013-03-22 15:29:02

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

5

Author

stevecheng (10074)

Entry type

Definition

Classification

msc 53A05

Related topic

FirstFundamentalForm

Related topic

ShapeOperator

Related topic

NormalSection

Related topic

NormalCurvatures

Defines

normal curvature

Defines

principal direction

Defines

principal curvature

Defines

Weingarten matrix

Defines

Weingarten map

Defines

Gauss map

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