Course Geometry

Coarse geometry
A metric space is a space equipped with a notion of distance. We use d(x, y)
to denote the distance from the point x to the point y. In topology, we study
metric spaces mainly by considering the continuous maps between them.
When defining continuity of a map, we look only at very small distances.


We say that a map f : X → Y is uniformly continuous if for all > 0 there
exists some δ > 0 such that whenever d(x, y) < δ we have d(f (x), f (y)) < .
In contrast, coarse geometry considers the larger scale structure of a
space. A map f : X → Y is said to be coarse if
• for all R > 0 there exists some S > 0 such that whenever d(x, y) < R
we have d(f (x), f (y)) < S, and
• whenever a subset B of Y is bounded, its preimage f −1 (B) is bounded
in X.
In topology, we often consider spaces up to homeomorphism. Two spaces
X and Y are homeomorphic if there is a continuous map f : X → Y which
has a continuous inverse f −1 : Y → X. In coarse geometry, we consider
spaces up to coarse equivalence. The spaces X and Y are considered coarsely
equivalent if there are coarse maps f : X → Y and g : Y → X which are
inverses up to a uniformly bounded error. For example, the real line R
and the space of integers Z are coarsely equivalent. Note that R and Z
are very different when considering their small scale structure, but could be
considered the same at a large scale.


Scalar curvature

An n-dimensional manifold is a space which locally resembles the Euclidean
space Rn . For example, a sphere in three-dimensional space locally resembles
the plane R2 . To illustrate this example, consider the surface of the Earth.
The Earth is (homeomorphic to) a sphere but, when looked at locally, seems
to be flat, like the plane R2 .
A Riemannian metric on a manifold is a notion of both distance and
angle. A manifold equipped with a Riemannian metric is called a Riemannian manifold. When a manifold is equipped with such information, we can
define a...