Hello. This is the B3.2 geometry of surfaces lecture one.
So my primary reference for this course will be the Hitchin online lecture notes.
There's various notes by Ritter, Segal, Earl and so on, which are also helpful
And there are lots of books available. So, surfaces are two dimensional objects, in pictures
Here is the 2 sphere, here is the two dimensional Torus,
This would be a surface of genius two, and so there are many different kinds of surfaces.
You can make surfaces out of Plasticine or Play-Dough.
So this course we'll discuss three different kinds of surfaces.
Firstly, topological surfaces, which are special examples of topological spaces.
This will be covered in lectures one to five, and in Section two of the Hitchin lecture notes.
Some highlights of our treatment of topological surfaces will be the Euler characteristic
of a topological surface, The notion of orientability;
a surface can be orientable or not orientable, And the classification of compact surfaces;
it turns out that if you know the Euler characteristic of the surface
and you know whether it's orientable or not, then you know what the surface is, up to isomorphism.
The second notion of surface is Riemann surfaces. So these are a generalisation of complex analysis
and are surfaces which have a notion of holomorphic function.
So Riemann surfaces are topological spaces, together with
an extra geometric structure, a structure
which tells you which complex functions on surfaces are holomorphic, broadly.
And they'll be the subject of lectures five to eight and Hitchin notes section three.
Some highlights of the Riemann surfaces part of the course is the Riemann-Hurwitz formula.
The third kind of surface we're going to talk about are smooth surfaces,
which may or may not be embedded in R 3, three dimensional space,
as the surface of some object, for example. We can also consider abstract surfaces
which don't come with an embedding in three dimensional space.
This is a topic of the last half of the course lectures 9 through to 16
and sections four and sections five.
So, a smooth surface is also a topological space with an additional geometric structure,
One which tells you which functions on the surface 27
but we also consider extra geometric structures on top of the smooth surface,
namely a second fundamental form or Riemannian metric.
And some highlights of this part of the course are the notion of Gaussian curvature;
That's a function you attach to a Riemannian metric, the Gauss-Bonnet theorem, geodesics
and hyperbolic geometry. So these are all part of Riemannian geometry,
which is an area of geometry which studies two dimensional
Surfaces or high dimensional surfaces, which are called manifolds,
together with a notion of distance. So this course is a little bit
like a tasting menu in a restaurant, It gives you a little bit of lots of subjects.
So we don't really do anything properly in depth, And the subjects which this is hinting at
are actually vast And you can spend years learning of them.
And I won't have time to say a lot of things I would like to, and if you want to do the course
Well, then you should probably do some extra reading. So, for example, topological surfaces,
You could go off in the direction of topological manifolds
or Euler characteristic is a part of the subject Algebraic topology. Riemann surfaces is really
a one dimensional case of complex geometry, which is again a huge subject.
Smooth surfaces and Riemannian metrics On them is the beginnings of differential geometry
and Riemannian geometry, which again, are very big subjects.
OK, so let's move on to section two, which will be about topological surfaces.
We'll begin the section 2.1 on the background from topology. So that, A5 topology is essential
For this course. If you don't know about topoogy then I suggest you
Pause at this point and go away and find out; one of my favourite books,
undergraduate texts on Topology, would be the book by Wilson Sutherland
on Introduction to Topological Spaces. So let's just remember that a topological
space is a set X together with a collection curly T of subsets of X,
These subsets are called open sets in X. They have to have three properties,
Firstly, the empty set is open and the entirety of X is open.
Secondly, if you have any two open sets, U and V then the interstection U intersection V, is in
T, the intersection of open spaces is open. By induction,
That means that the intersection of a finite number of open sets
is open. Thirdly, if you have
Possibly infinitely many open sets in T parametrised by some indexing set big I, then
the union of those open sets is also open. So finite intersections of open sets are open,
but arbitrary Unions of open sets are open.
So a map f going between X and Y, two topological spaces is called continuous
If, whenever V is an open set in Y, then the pullback f inverse of V is an open set in X.
A topological space Is called Hausdorff if whenever two
points x and y are in big X which are different x is not equal to y, then you can find open sets
U containing x and V containing y with U intersection V is the empty set.
A topological space X is called compact If every open cover of X has a finite sub-cover,
so being compact kind of means that there aren't any infinite ends.
So, for example, R would not be compact because you can go to infinity without -
So another form of compactness is that every sequence should have a subsequence
which has a limit. A function f going from X into Y is a homeomorphism
If it is continuous and invertible with continuous inverse.
So homeomoprhisms are isomorphisms of topological spaces.
And finally, if X is a topological space, Y is contained in X as a subset,
then the subspace topology of Y is T Y,
which is the intersection Y intersect U for all open sets in X and, implicitly,
whenever we define something as a subspace, for example, if we define the two sphere as
the set of points of radius one in R three, if we define something as a subset of
topological space, then implicitly we give it the subspace topology.
Perhaps one other thing I could add is this notion of when a topological space is second countable.
That means the topology has a countable basis where a basis for the topological space is
A set of open sets such that every other open set is a union of finite
intersections of sets in the basis. So we need to be familiar with these
ideas of topological spaces and with the basic constructions,
which are done in topology such as making, forming a quotient topological space.
OK, so now let's define the topological surfaces. So a topologicalk surface or in this
part of the court we'll often just refer to them as surfaces, a surface X is a Hausdorff
topological space such that each point little x in big X
has an open neighbourhood U together with a homeomorphism phi going from U into V
to an open subset V contained in R 2. So the triple U and V and phi is called
a chart on X. So charts are not going to play a particularly important role in this first part
But then later on when we talk about Riemann surfaces or smooth surfaces,
the definition is going to make very heavy use of this notion of charts.
So, a few remarks, firstly, in older books, a surface is called closed
If it is compact, so I'm not going to use
that term because I feel like that can get confused with the notion of a closed set.
Secondly, it's better to also require your surface X to be either second countable or paracompact;
Second countable is actually the better notion, it's slightly stronger in its context.
These are both global topological conditions
but they are automatic for compact surfaces, which is what we mostly care about,
So for simplicity, we're not going to worry about second countability or paracompactness,
But if you go and read the definition of smooth manifolds, let's say in a book about
differential geometry, then hopefully it will
include the notion of second countability, which excludes
Some non-compact surfaces or whatever, which are in some sense very, very big.
Thirdly, we can generalise this definition of topological surfaces to define the
notion of topological manifolds of dimension n. So we could define a topological manifold to be
a Hausdorff topological space - Ideally, we'd also define it
required to be second countable - which is locally homeomoprhic to R to the n.
That is, each point in X has an open neighbourhood which is homeomorphicto an open set in R to the n.
So then surfaces as we would define them, are just topological manifolds
Of dimension two. So we'll see later that compact surfaces can be completely classified.
You can understand them very explicitly, however, in dimensions n greater than two,
classifying compact topological n manifolds Is very difficult. Actually, in some sense
it's most difficult in dimensions Three and four, and then,
after dimension five, it starts to get easier, But even so, the answer is very, very complicated.
So two dimensions are uncharacteristically simple in
The world of topological manifolds. OK. Let's look at some examples. Firstly,
any open subset X of R 2 is a topological surface which you can cover by a single chart
U, V, phi in which U and V are both equal to X and phi is the identity map.
The second example, let's think about the two sphere,
S 2, which is the set of points x and y and z in three dimensional space
R 3 for which x squared plus y squared plus z squared equals one.
So implicitly, we mean that R 3 is a topological space with the usual topology,
and then S 2 has the subset topology,
that is open subsets of S 2 are the intersections S 2 with open subsets of R three.
Now we can cover the two sphere by two charts U 1, V 1, phi 1, U 2, V 2, phi 2.
So let's take U 1 to be the 2 sphere take away the south pole,
0,0,minus 1. U 2 to be S 2 take away the north pole,
0, 0, 1, those points
in the intersection of the Z axis with the 2 sphere. Take V 1 and V 2 to be R squared,
And the maps phi i, identifying U i with V i, are given by phi 1 of x comma y comma z
Is the point x, y divided by one plus z. So that obviously has a pole
when z is minus one, which is why we exclude the point
0, 0, minus 1 and phi 2 of x, y, z
Is x,y divided by one minus z, that again has a pole at z equals one to exclude the point 0, 0, 1.
So you can't cover two sphere with one chart, you need at least two. For more complicated surfaces,
You may need many charts to cover the entirety of your surface.
Okay. So S 2 is locally homeomorphic to R squared, it is Hausdorff
And also second countable
because it's a subspace of R 3, which is both Hausdorff and second countable,
And Hausdorff and second countable are inherited by subspaces so therefore S 2 is a topological
surface and it's also compact by the Heine-Borel theorem because it is closed and bounded in R 3.