4Hyperbolic geometry
IB Geometry
4.2 Riemannian metrics
Finally, we get to the idea of a Riemannian metric. The basic idea of a Rieman-
nian metric is not too unfamiliar. Presumably, we have all seen maps of the
Earth, where we try to draw the spherical Earth on a piece of paper, i.e. a subset
of
R
2
. However, this does not behave like
R
2
. You cannot measure distances on
Earth by placing a ruler on the map, since distances are distorted. Instead, you
have to find the coordinates of the points (e.g. the longitude and latitude), and
then plug them into some complicated formula. Similarly, straight lines on the
map are not really straight (spherical) lines on Earth.
We really should not think of Earth a subset of
R
2
. All we have done was
to “force” Earth to live in
R
2
to get a convenient way of depicting the Earth, as
well as a convenient system of labelling points (in many map projections, the
x
and y axes are the longitude and latitude).
This is the idea of a Riemannian metric. To describe some complicated
surface, we take a subset
U
of
R
2
, and define a new way of measuring distances,
angles and areas on
U
. All these information are packed into an entity known
as the Riemannian metric.
Definition (Riemannian metric). We use coordinates (
u, v
)
∈ R
2
. We let
V ⊆ R
2
be open. Then a Riemannian metric on
V
is defined by giving
C
∞
functions E, F, G : V → R such that
E(P ) F (P )
F (P ) G(P )
is a positive definite definite matrix for all P ∈ V .
Alternatively, this is a smooth function that gives a 2
×
2 symmetric positive
definite matrix, i.e. inner product
h·, ·i
P
, for each point in
V
. By definition, if
e
1
, e
2
are the standard basis, then
he
1
, e
1
i
P
= E(P )
he
1
, e
2
i
P
= F (P )
he
2
, e
2
i
P
= G(P ).
Example. We can pick
E
=
G
= 1 and
F
= 0. Then this is just the standard
Euclidean inner product.
As mentioned, we should not imagine
V
as a subset of
R
2
. Instead, we should
think of it as an abstract two-dimensional surface, with some coordinate system
given by a subset of
R
2
. However, this coordinate system is just a convenient way
of labelling points. They do not represent any notion of distance. For example,
(0, 1) need not be closer to (0, 2) than to (7, 0). These are just abstract labels.
With this in mind,
V
does not have any intrinsic notion of distances, angles
and areas. However, we do want these notions. We can certainly write down
things like the difference of two points, or even the compute the derivative of
a function. However, these numbers you get are not meaningful, since we can
easily use a different coordinate system (e.g. by scaling the axes) and get a
different number. They have to be interpreted with the Riemannian metric.
This tells us how to measure these things, via an inner product “that varies with
space”. This variation in space is not an oddity arising from us not being able to
make up our minds. This is since we have “forced” our space to lie in
R
2
. Inside
V
, going from (0
,
1) to (0
,
2) might be very different from going from (5
,
5) to
(6
,
5), since coordinates don’t mean anything. Hence our inner product needs
to measure “going from (0
,
1) to (0
,
2)” differently from “going from (5
,
5) to
(6, 5)”, and must vary with space.
We’ll soon come to defining how this inner product gives rise to the notion
of distance and similar stuff. Before that, we want to understand what we can
put into the inner product
h·, ·i
P
. Obviously these would be vectors in
R
2
, but
where do these vectors come from? What are they supposed to represent?
The answer is “directions” (more formally, tangent vectors). For example,
h
e
1
,
e
1
i
P
will tell us how far we actually are going if we move in the direction
of e
1
from
P
. Note that we say “move in the direction of e
1
”, not “move by
e
1
”. We really should read this as “if we move by
h
e
1
for some small
h
, then
the distance covered is
h
p
he
1
, e
1
i
P
”. This statement is to be interpreted along
the same lines as “if we vary
x
by some small
h
, then the value of
f
will vary
by
f
0
(
x
)
h
”. Notice how the inner product allows us to translate a length in
R
2
(namely khe
1
k
eucl
= h) into the actual length in V .
What we needed for this is just the norm induced by the inner product. Since
what we have is the whole inner product, we in fact can define more interesting
things such as areas and angles. We will formalize these ideas very soon, after
getting some more notation out of the way.
Often, instead of specifying the three functions separately, we write the metric
as
E du
2
+ 2F du dv + G dv
2
.
This notation has some mathematical meaning. We can view the coordinates
as smooth functions
u
:
V → R
,
v
:
U → R
. Since they are smooth, they have
derivatives. They are linear maps
du
P
: R
2
→ R dv
P
: R
2
→ R
(h
1
, h
2
) 7→ h
1
(h
1
, h
2
) 7→ h
2
.
These formula are valid for all
P ∈ V
. So we just write d
u
and d
v
instead.
Since they are maps
R
2
→ R
, we can view them as vectors in the dual space,
d
u,
d
v ∈
(
R
2
)
∗
. Moreover, they form a basis for the dual space. In particular,
they are the dual basis to the standard basis e
1
, e
2
of R
2
.
Then we can consider d
u
2
,
d
u
d
v
and d
v
2
as bilinear forms on
R
2
. For
example,
du
2
(h, k) = du(h)du(k)
du dv(h, k) =
1
2
(du(h)dv(k) + du(k)dv(h))
dv
2
(h, k) = dv(h)dv(k)
These have matrices
1 0
0 0
,
0
1
2
1
2
0
,
0 0
0 1
respectively. Then we indeed have
E du
2
+ 2F du dv + G dv
2
=
E F
F G
.
We can now start talking about what this is good for. In standard Euclidean
space, we have a notion of length and area. A Riemannian metric also gives a
notion of length and area.
Definition (Length). The length of a smooth curve
γ
= (
γ
1
, γ
2
) : [0
,
1]
→ V
is
defined as
Z
1
0
E ˙γ
2
1
+ 2F ˙γ
1
˙γ
2
+ G ˙γ
2
2
1
2
dt,
where E = E(γ
1
(t), γ
2
(t)) etc. We can also write this as
Z
1
0
h˙γ, ˙γi
1
2
γ(t)
dt.
Definition (Area). The area of a region W ⊆ V is defined as
Z
W
(EG − F
2
)
1
2
du dv
when this integral exists.
In the area formula, what we are integrating is just the determinant of the
metric. This is also known as the Gram determinant.
We define the distance between two points
P
and
Q
to be the infimum of
the lengths of all curves from
P
to
Q
. It is an exercise on the second example
sheet to prove that this is indeed a metric.
Example. We will not do this in full detail — the details are to be filled in in
the third example sheet.
Let V = R
2
, and define the Riemannian metric by
4(du
2
+ dv
2
)
(1 + u
2
+ v
2
)
2
.
This looks somewhat arbitrary, but we shall see this actually makes sense by
identifying
R
2
with the sphere by the stereographic projection
π
:
S
2
\{N} → R
2
.
For every point
P ∈ S
2
, the tangent plane to
S
2
at
P
is given by
{
x
∈ R
3
:
x
·
−−→
OP
= 0
}
. Note that we translated it so that
P
is the origin, so that we can
view it as a vector space (points on the tangent plane are points “from
P
”).
Now given any two tangent vectors x
1
,
x
2
⊥
−−→
OP
, we can take the inner product
x
1
· x
2
in R
3
.
We want to say this inner product is “the same as” the inner product provided
by the Riemannian metric on R
2
. We cannot just require
x
1
· x
2
= hx
1
, x
2
i
π(P )
,
since this makes no sense at all. Apart from the obvious problem that x
1
,
x
2
have three components but the Riemannian metric takes in vectors of two
components, we know that x
1
and x
2
are vectors tangent to
P ∈ S
2
, but to
apply the Riemannian metric, we need the corresponding tangent vector at
π(P ) ∈ R
2
. To do so, we act by dπ
p
. So what we want is
x
1
· x
2
= hdπ
P
(x
1
), dπ
P
(x
2
)i
π(P )
.
Verification of this equality is left as an exercise on the third example sheet. It
is helpful to notice
π
−1
(u, v) =
(2u, 2v, u
2
+ v
2
− 1)
1 + u
2
+ v
2
.
In some sense, we say the surface
S
2
\ {N}
is “isometric” to
R
2
via the
stereographic projection
π
. We can define the notion of isometry between two
open sets with Riemannian metrics in general.
Definition (Isometry). Let
V,
˜
V ⊆ R
2
be open sets endowed with Riemannian
metrics, denoted as h·, ·i
P
and h·, ·i
∼
Q
for P ∈ V, Q ∈
˜
V respectively.
A diffeomorphism (i.e.
C
∞
map with
C
∞
inverse)
ϕ
:
V →
˜
V
is an isometry
if for every P ∈ V and x, y ∈ R
2
, we get
hx, yi
P
= hdϕ
P
(x), dϕ
P
(y)i
∼
ϕ(P )
.
Again, in the definition, x and y represent tangent vectors at
P ∈ V
, and
on the right of the equality, we need to apply d
ϕ
P
to get tangent vectors at
ϕ(P ) ∈
˜
V .
How are we sure this indeed is the right definition? We, at the very least,
would expect isometries to preserve lengths. Let’s see this is indeed the case. If
γ
: [0
,
1]
→ V
is a
C
∞
curve, the composition
˜γ
=
ϕ ◦ γ
: [0
,
1]
→
˜
V
is a path in
˜
V . We let P = γ(t), and hence ϕ(P ) = ˜γ(t). Then
h˜γ
0
(t), ˜γ
0
(t)i
∼
˜γ(t)
= hdϕ
P
◦ γ
0
(t), dϕ
P
◦ γ
0
(t)i
∼
ϕ(P )
= hγ
0
(t), γ
0
(t)i
γ(t)=P
.
Integrating, we obtain
length(˜γ) = length(γ) =
Z
1
0
hγ
0
(t), γ
0
(t)i
γ(t)
dt.