5Smooth embedded surfaces (in ℝ3)
IB Geometry
5.1 Smooth embedded surfaces
So far, we have been studying some specific geometries, namely Euclidean, spheri-
cal and hyperbolic geometry. From now on, we go towards greater generality, and
study arbitrary surfaces. We will mostly work with surfaces that are smoothly
embedded as subsets of
R
3
, we can develop notions parallel to those we have had
before, such as Riemannian metrics and lengths. At the very end of the course,
we will move away from the needless restriction restriction that the surface is
embedded in R
3
, and study surfaces that just are.
Definition (Smooth embedded surface). A set
S ⊆ R
3
is a (parametrized)
smooth embedded surface if every point
P ∈ S
has an open neighbourhood
U ⊆ S
(with the subspace topology on
S ⊆ R
3
) and a map
σ
:
V → U
from an open
V ⊆ R
2
to U such that if we write σ(u, v) = (x(u, v), y(u, v), z(u, v)), then
(i) σ is a homeomorphism (i.e. a bijection with continuous inverse)
(ii) σ
is
C
∞
(smooth) on
V
(i.e. has continuous partial derivatives of all orders).
(iii)
For all
Q ∈ V
, the partial derivatives
σ
u
(
Q
) and
σ
v
(
Q
) are linearly inde-
pendent.
Recall that
σ
u
(Q) =
∂σ
∂u
(Q) =
∂x
∂u
∂y
∂u
∂z
∂u
(Q) = dσ
Q
(e
1
),
where e
1
, e
2
is the standard basis of R
2
. Similarly, we have
σ
v
(Q) = dσ
Q
(e
2
).
We define some terminology.
Definition (Smooth coordinates). We say (
u, v
) are smooth coordinates on
U ⊆ S.
Definition (Tangent space). The subspace of
R
3
spanned by
σ
u
(
Q
)
, σ
v
(
Q
) is
the tangent space T
P
S to S at P = σ(Q).
Definition (Smooth parametrisation). The function
σ
is a smooth parametrisa-
tion of U ⊆ S.
Definition (Chart). The function σ
−1
: U → V is a chart of U.
Proposition. Let
σ
:
V → U
and
˜σ
:
˜
V → U
be two
C
∞
parametrisations of a
surface. Then the homeomorphism
ϕ = σ
−1
◦ ˜σ :
˜
V → V
is in fact a diffeomorphism.
This proposition says any two parametrizations of the same surface are
compatible.
Proof.
Since differentiability is a local property, it suffices to consider
ϕ
on some
small neighbourhood of a point in
V
. Pick our favorite point (
v
0
, u
0
)
∈
˜
V
. We
know σ = σ(u, v) is differentiable. So it has a Jacobian matrix
x
u
x
v
y
u
y
v
z
y
z
v
.
By definition, this matrix has rank two at each point. wlog, we assume the first
two rows are linearly independent. So
det
x
u
x
v
y
u
y
v
6= 0
at (v
0
, u
0
) ∈
˜
V . We define a new function
F (u, v) =
x(u, v)
y(u, v)
.
Now the inverse function theorem applies. So
F
has a local
C
∞
inverse, i.e.
there are two open neighbourhoods (
u
0
, v
0
)
∈ N
and
F
(
u
0
, v
0
)
∈ N
0
⊆ R
2
such
that f : N → N
0
is a diffeomorphism.
Writing
π
:
˜σ → N
0
for the projection
π
(
x, y, z
) = (
x, y
) we can put these
things in a commutative diagram:
σ(N)
N N
0
π
F
σ
.
We now let
˜
N
=
˜σ
−1
(
σ
(
N
)) and
˜
F
=
π ◦ ˜σ
, which is yet again smooth. Then we
have the following larger commutative diagram.
σ(N)
N N
0
˜
N
π
F
σ
˜
F
˜σ
.
Then we have
ϕ = σ
−1
◦ ˜σ = σ
−1
◦ π
−1
◦ π ◦ ˜σ = F
−1
◦
˜
F ,
which is smooth, since
F
−1
and
˜
F
are. Hence
ϕ
is smooth everywhere. By
symmetry of the argument,
ϕ
−1
is smooth as well. So this is a diffeomorphism.
A more practical result is the following:
Corollary. The tangent plane T
Q
S is independent of parametrization.
Proof. We know
˜σ(˜u, ˜v) = σ(ϕ
1
(˜u, ˜v), ϕ
2
(˜u, ˜v)).
We can then compute the partial derivatives as
˜σ
˜u
= ϕ
1,˜u
σ
u
+ ϕ
2,˜u
σ
v
˜σ
˜v
= ϕ
1,˜v
σ
u
+ ϕ
2,˜v
σ
v
Here the transformation is related by the Jacobian matrix
ϕ
1,˜u
ϕ
1,˜v
ϕ
2,˜u
ϕ
2,˜v
= J(ϕ).
This is invertible since
ϕ
is a diffeomorphism. So (
σ
˜u
, σ
˜v
) and (
σ
u
, σ
v
) are
different basis of the same two-dimensional vector space. So done.
Note that we have
˜σ
˜u
× ˜σ
˜v
= det(J(ϕ))σ
u
× σ
v
.
So we can define
Definition (Unit normal). The unit normal to S at Q ∈ S is
N = N
Q
=
σ
u
× σ
v
kσ
u
× σ
v
k
,
which is well-defined up to a sign.
Often, instead of a parametrization
σ
:
V ⊆ R
2
→ U ⊆ S
, we want the
function the other way round. We call this a chart.
Definition (Chart). Let
S ⊆ R
3
be an embedded surface. The map
θ
=
σ
−1
:
U ⊆ S → V ⊆ R
2
is a chart.
Example. Let
S
2
⊆ R
3
be a sphere. The two stereographic projections from
±e
3
give two charts, whose domain together cover S
2
.
Similar to what we did to the sphere, given a chart
θ
:
U → V ⊆ R
2
, we can
induce a Riemannian metric on
V
. We first get an inner product on the tangent
space as follows:
Definition (First fundamental form). If
S ⊆ R
3
is an embedded surface, then
each
T
Q
S
for
Q ∈ S
has an inner product from
R
3
, i.e. we have a family of inner
products, one for each point. We call this family the first fundamental form.
This is a theoretical entity, and is more easily worked with when we have
a chart. Suppose we have a parametrization
σ
:
V → U ⊆ S
, a
,
b
∈ R
2
, and
P ∈ V . We can then define
ha, bi
P
= hdσ
P
(a), dσ
P
(b)i
R
3
.
With respect to the standard basis e
1
,
e
2
∈ R
2
, we can write the first fundamental
form as
E du
2
+ 2F du dv + G dv
2
,
where
E = hσ
u
, σ
u
i = he
1
, e
1
i
P
F = hσ
u
, σ
v
i = he
1
, e
2
i
P
G = hσ
v
, σ
v
i = he
2
, e
2
i
P
.
Thus, this induces a Riemannian metric on
V
. This is also called the first
fundamental form corresponding to
σ
. This is what we do in practical examples.
We will assume the following property, which we are not bothered to prove.
Proposition. If we have two parametrizations related by
˜σ
=
σ ◦ ϕ
:
˜
V → U
,
then ϕ :
˜
V → V is an isometry of Riemannian metrics (on V and
˜
V ).
Definition (Length and energy of curve). Given a smooth curve Γ : [
a, b
]
→
S ⊆ R
3
, the length of γ is
length(Γ) =
Z
b
a
kΓ
0
(t)k dt.
The energy of the curve is
energy(Γ) =
Z
b
a
kΓ
0
(t)k
2
dt.
We can think of the energy as something like the kinetic energy of a particle
along the path, except that we are missing the factor of
1
2
m
, because they are
annoying.
How does this work with parametrizations? For the sake of simplicity, we
assume Γ([
a, b
])
⊆ U
for some parametrization
σ
:
V → U
. Then we define the
new curve
γ = σ
−1
◦ Γ : [a, b] → V.
This curve has two components, say γ = (γ
1
, γ
2
). Then we have
Γ
0
(t) = (dσ)
γ(t)
( ˙γ
1
(t)e
1
+ ˙γ
2
(t)e
2
) = ˙γ
1
σ
u
+ ˙γ
2
σ
v
,
and thus
kΓ
0
(t)k = h˙γ, ˙γi
1
2
P
= (E ˙γ
2
1
+ 2F ˙γ
1
˙γ
2
+ G ˙γ
2
2
)
1
2
.
So we get
length Γ =
Z
b
a
(E ˙γ
2
1
+ 2F ˙γ
1
˙γ
2
+ G ˙γ
2
2
)
1
2
dt.
Similarly, the energy is given by
energy Γ =
Z
b
a
(E ˙γ
2
1
+ 2F ˙γ
1
˙γ
2
+ G ˙γ
2
2
) dt.
This agrees with what we’ve had for Riemannian metrics.
Definition (Area). Given a smooth
C
∞
parametrization
σ
:
V → U ⊆ S ⊆ R
3
,
and a region T ⊆ U, we define the area of T to be
area(T ) =
Z
θ(T )
p
EG − F
2
du dv,
whenever the integral exists (where θ = σ
−1
is a chart).
Proposition. The area of
T
is independent of the choice of parametrization.
So it extends to more general subsets
T ⊆ S
, not necessarily living in the image
of a parametrization.
Proof. Exercise!
Note that in examples,
σ
(
V
) =
U
often is a dense set in
S
. For example, if
we work with the sphere, we can easily parametrize everything but the poles. In
that case, it suffices to use just one parametrization σ for area(S).
Note also that areas are invariant under isometries.