6Abstract smooth surfaces

IB Geometry



6 Abstract smooth surfaces
While embedded surfaces are quite general surfaces, they do not include, say,
the hyperbolic plane. We can generalize our notions by considering surfaces
“without embedding in R
3
”. These are known as abstract surfaces.
Definition (Abstract smooth surface). An abstract smooth surface
S
is a metric
space (or Hausdorff (and second-countable) topological space) equipped with
homeomorphisms
θ
i
:
U
i
V
i
, where
U
i
S
and
V
i
R
2
are open sets such
that
(i) S =
S
i
U
i
(ii) For any i, j, the transition map
φ
ij
= θ
j
θ
1
i
: θ
j
(U
i
U
j
) θ
i
(U
i
U
j
)
is a diffeomorphism. Note that
θ
j
(
U
i
U
j
) and
θ
i
(
U
i
U
j
) are open
sets in
R
2
. So it makes sense to talk about whether the function is a
diffeomorphism.
Like for embedded surfaces, the maps
θ
i
are called charts, and the collection
of θ
i
’s satisfying our conditions is an atlas etc.
Definition (Riemannian metric on abstract surface). A Riemannian metric on
an abstract surface is given by Riemannian metrics on each
V
i
=
θ
i
(
U
i
) subject to
the compatibility condition that for all
i, j
, the transition map
φ
ij
is an isometry,
i.e.
h
P
(a), dϕ
P
(b)i
ϕ(P )
= ha, bi
P
Note that on the left, we are computing the Riemannian metric on
V
i
, while on
the left, we are computing it on V
j
.
Then we can define lengths, areas, energies on an abstract surface S.
It is clear that every embedded surface is an abstract surface, by forgetting
that it is embedded in R
3
.
Example. The three classical geometries are all abstract surfaces.
(i) The Euclidean space R
2
with dx
2
+ dy
2
is an abstract surface.
(ii)
The sphere
S
2
R
2
, being an embedded surface, is an abstract surface
with metric
4(dx
2
+ dy
2
)
(1 + x
2
+ y
2
)
2
.
(iii) The hyperbolic disc D R
2
is an abstract surface with metric
4(dx
2
+ dy
2
)
(1 x
2
y
2
)
2
.
and this is isometric to the upper half plane H with metric
dx
2
+ dy
2
y
2
Note that in the first and last example, it was sufficient to use just one chart
to cover every point of the surface, but not for the sphere. Also, in the case of the
hyperbolic plane, we can have many different charts, and they are compatible.
Finally, we notice that we really need the notion of abstract surface for the
hyperbolic plane, since it cannot be realized as an embedded surface in
R
3
. The
proof is not obvious at all, and is a theorem of Hilbert.
One important thing we can do is to study the curvature of surfaces.
Given a
P S
, the Riemannian metric (on a chart) around
P
determines
a “reparametrization” by geodesics, similar to embedded surfaces. Then the
metric takes the form
dρ
2
+ G(ρ, θ) dθ
2
.
We then define the curvature as
K =
(
G)
ρρ
G
.
Note that for embedded surfaces, we obtained this formula as a theorem. For
abstract surfaces, we take this as a definition.
We can check how this works in some familiar examples.
Example.
(i) In R
2
, we use the usual polar coordinates (ρ, θ), and the metric becomes
dρ
2
+ ρ
2
dθ
2
,
where x = ρ cos θ and y = ρ sin θ. So the curvature is
(
G)
ρρ
G
=
(ρ)
ρρ
ρ
= 0.
So the Euclidean space has zero curvature.
(ii)
For the sphere
S
, we use the spherical coordinates, fixing the radius to be
1. So we specify each point by
σ(ρ, θ) = (sin ρ cos θ, sin ρ sin θ, cos ρ).
Note that
ρ
is not really the radius in spherical coordinates, but just one
of the angle coordinates. We then have the metric
dρ
2
+ sin
2
ρ dθ
2
.
Then we get
G = sin ρ,
and K = 1.
(iii)
For the hyperbolic plane, we use the disk model
D
, and we first express
our original metric in polar coordinates of the Euclidean plane to get
2
1 r
2
2
(dr
2
+ r
2
dθ
2
).
This is not geodesic polar coordinates, since
r
is given by the Euclidean
distance, not hyperbolic distance. We will need to put
ρ = 2 tanh
1
r, dρ =
2
1 r
2
dr.
Then we have
r = tanh
ρ
2
,
which gives
4r
2
(1 r
2
)
2
= sinh
2
ρ.
So we finally get
G = sinh ρ,
with
K = 1.
We see that the three classic geometries are characterized by having constant
0, 1 and 1 curvatures.
We are almost able to state the Gauss-Bonnet theorem. Before that, we need
the notion of triangulations. We notice that our old definition makes sense for
(compact) abstract surfaces
S
. So we just use the same definition. We then
define the Euler number of an abstract surface as
e(S) = F E + V,
as before. Assuming that the Euler number is independent of triangulations, we
know that this is invariant under homeomorphisms.
Theorem (Gauss-Bonnet theorem). If the sides of a triangle
ABC S
are
geodesic segments, then
Z
ABC
K dA = (α + β + γ) π,
where
α, β, γ
are the angles of the triangle, and d
A
is the “area element” given
by
dA =
p
EG F
2
du dv,
on each domain
U S
of a chart, with
E, F, G
as in the respective first
fundamental form.
Moreover, if S is a compact surface, then
Z
S
K dA = 2πe(S).
We will not prove this theorem, but we will make some remarks. Note that
we can deduce the second part from the first part. The basic idea is to take a
triangulation of
S
, and then use things like each edge belongs to two triangles
and each triangle has three edges.
This is a genuine generalization of what we previously had for the sphere
and hyperbolic plane, as one can easily see.
Using the Gauss-Bonnet theorem, we can define the curvature
K
(
P
) for a
point
P S
alternatively by considering triangles containing
P
, and then taking
the limit
lim
area0
(α + β + γ) π
area
= K(P ).
Finally, we note how this relates to the problem of the parallel postulate we have
mentioned previously. The parallel postulate, in some form, states that given a
line and a point not on it, there is a unique line through the point and parallel
to the line. This holds in Euclidean geometry, but not hyperbolic and spherical
geometry.
It is a fact that this is equivalent to the axiom that the angles of a triangle
sum to
π
. Thus, the Gauss-Bonnet theorem tells us the parallel postulate is
captured by the fact that the curvature of the Euclidean plane is zero everywhere.