Shapes of Space
Dan Knopf
February 13, 2008
Mathematicians, like biologists, like to classify things. One of the major
areas of research in topology is classifying smooth manifolds.
That begs the question ”What is a manifold?”. A manifold is a space that
locally looks like Euclidean space. A smooth manifold is a manifold you can do
calculus. We say that two smooth manifolds are the same if we can deform one
into the other smoothly without tearing them.
In dimension one the only two manifolds are the real line and circle. Knots
in three space are just circles because you can move into higher dimensions and
unknot them.
Medieval Europeans thought that the world was flat. In reality we knot
that the world is sphere but locally you can’t tell the difference. From local
information it is impossible to tell that we don’t live on a bagel.
What are all the possible two manifolds? We have the plane, spheres, and
bagels with any number of holes. Is this a complete list? Consider the Möbius
band. It looks locally like the plane everywhere except for around the boundary
(Which is a circle). So why not turn it into a manifold by gluing a disk (which
also has a circle as a boundary) to the boundary of the Möbius band. You can’t
perform this procedure in three dimensions but mathematically it works out
fine. This is the real projective plane. It is a manifolds and it is nonorientable.
By the techniques of algebraic topology you can show that any compact
connected orientable manifold is either a sphere, or a bagel with some number
of holes. You can proof this by showing that the Euler characteristic χ(M 2 ) =
2 − 2g (where g is the genus) or number of holes) is an invariant.
Unfortunately Dr. Knopf is not a topologist. He is a geometer and geometers
measure things. This begs the questions ”What do you men when you talk
about the shape of a surface and can you classify a surface by its shape?”. Well
shape depends on angle and distance so we say that the shape of a surface is
encapsulated in a Riemannian metric (a familiar example is the dot product in
Rm ). On a surface this metric has to change as you move about the space.
Can we make a torus look locally geometrically like the plane? Intuitively
the answer should be no, because the surface of a torus has curvature. What is
curvature? One way to observe its effect is to think of triangles on the surface
as opposed in the plane. On a sphere the angles of the triangle are greater than
you’d expect. On a hyperboloid the angles are less than you’d expect.
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Gauss discovered that he could measure curvature as part of his Theorem
Egregium. His method was considering a tangent plane to the surface. If all
curves on the surface lie on one side of the plane curvature is positive. If all
curves are in the plane it has negative curvature. Finally if some are on one
side and some are on the other the curvature is negative. Curvature is denoted
by the letter R for Riemann.
A torus exhibits all kinds of curvature at different points on the surface.
Therefore the curvature is a function of position. It turns out that by integrating the curvature over the whole manifold you get a topological invariant.
Unfortunately this classification is very coarse.
The Uniformization theorem. You can deform any space so that its curvature
is constant everywhere. This gives a sort of canonical shape for any space. For
example there is a flat torus in dimension four.
Now lets think of three manifolds. How hard could that be? Classification
of two manifolds was completed in the 19th century. It turns out that to even
begin classifying three manifolds you need to solve the Poincare conjecture. Bill
Thurston posed his geometrization conjecture in the 1970s. This stated that you
can cut up any manifold into a bunch of pieces each of which has one of eight
different geometries. To prove this you write down a differential equation for
the Riemannian metric which makes it more homogeneous. Eventually as the
solution develops singularities you have to cut the manifold into pieces. When
the process is over you get a bunch of pieces each with its own geometry.
(Read the Shape of Space for more info)
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