About
Differential geometry is to look at the world using tools
from differential calculus. There are various motivations why we want
to look at such a frame work:
It appears as if nature likes geometry. Indeed, the two
most amazing theories, the theory of gravity as well as the theory of
the atomic world are both heavily based on ideas from geometry and in
particular differential geometry.
The motion of celestial bodies can be described as shortest paths in
a manifold. Space alone tells matter how it moves. On the other hand
matter bends space. In the very small, these rather complicated nonlinear
systems become linear. Still also the solutions of the fundamental quantum
mechanical equations can be seen as geodesics in a manifold.
Differential geometry studies Riemannian manifolds, spaces that look
locally Euclidean but where the inner product providing the
measuring stick can depend on the position. This is richer than
multi-variable calculus where the inner product is the usual dot product.
Riemannian manifolds can either be studied intrinsically on their own but
it is more convenient for applications to write a manifold as embedded in
an ambient Euclidean space. A theorem of John Nash shows that this is not
a loss of generality. For actual computations it can help however to have
explicit parametrizations of manifolds meaning having coordinates.
Our physical world around us is in first approximation a three dimensional
Euclidean space. Physical objects are manifolds with surface boundaries.
The surface boundary of an apple for example is a 2 dimensional manifold.
If the apple falls, its path is a curve in our 3 dimensional space.
IN this course, we try to be as concrete as possible. A prerequisite for
understanding differential geometry is not only multi-variable calculus
but also basic linear algebra.
A good approach to any learning is to give a goal which needs to be reached.
If pressed to give the most important three points to reach in one semester,
then this are 1) prove the fundamental theorem of curves in arbitrary dimensions,
2) a detailed and rigorous proof of the Gauss-Bonnet theorem for surfaces
and 3) build up the general frame work in arbitrary dimensions to understand the
two pillars of general relativity: geodesic motion and the Einstein equations.
