Course Outline

The course will be an introduction to the study of Differential Geometry---one of the more classical (and also one of the more appealing) subjects of modern mathematics. We will be primarily concerned with curves in the plane and in 3-space, and with surfaces in 3-space. These concepts are pretty much exactly what their names suggest, and the close connection between the mathematics and our intuition is part of the charm of the subject. Of course, that does not mean that we can depend on intuition rater than rigorous proof---we will give careful definitions and proofs as usual, but the fact is that our geometric intuition will often suggest the steps in a proof.

Another important reason that Differential Geometry has proved so interesting is that the objects it studies have been the ones most commonly used to model the real world. For example, Einstein's theory of General Relativity was modelled on a differential geometric theory developed half a century earlier by Bernhard Riemann, and the interplay between Physics and Differential Geometry has ever since been fruitful for pushing both fields forward. In recent years, one of the most promising models for a "Theory of Everything" has been "String Theory", which is based on objects that are basically curves that vibrate in different ways.

The course will be an introduction to the study of Differential Geometry---one of the more classical (and also one of the more appealing) subjects of modern mathematics. We will be primarily concerned with curves in the plane and in 3-space, and with surfaces in 3-space. These concepts are pretty much exactly what their names suggest, and the close connection between the mathematics and our intuition is part of the charm of the subject. Of course, that does not mean that we can depend on intuition rater than rigorous proof---we will give careful definitions and proofs as usual, but the fact is that our geometric intuition will often suggest the steps in a proof. Another important reason that Differential Geometry has proved so interesting is that the objects it studies have been the ones most commonly used to model the real world. For example, Einstein's theory of General Relativity was modelled on a differential geometric theory developed half a century earlier by Bernhard Riemann, and the interplay between Physics and Differential Geometry has ever since been fruitful for pushing both fields forward. In recent years, one of the most promising models for a "Theory of Everything" has been "String Theory", which is based on objects that are basically curves that vibrate in different ways.		Since the curves and surfaces that we will be dealing with "live" in the Euclidean spaces of two and three dimensions, we will start with a review of the basic geometry of these "ambient spaces". In fact, since it adds no extra difficulty, we will study the geometry of Euclidean spaces of arbitrary finite dimension, the so-called inner-product spaces that most have you have probably already come across in your linear algebra courses. (But even before that, we will start by taking a quick look at the very idea of Geometry---that is, we will try to answer the question, what is a geometric theory and what should our goals be in studying any kind of geometry.) It turns out that an essential tool for studying curved objects in geometry is the theory of ODE (ordinary differential equations) so we will give a brief review of that theory too. After this brief review of ODE and the "linear" geometry of Euclidean space we will commence on our study of the curved objects that live inside them. In fact, we will find that the notion of curvature plays a particularly central role in our study of both curves and surfaces (and indeed curvature turns out to be an important key to understanding higher dimensional geometric objects too).