# New Course! Differential Geometry!(Math 4020) Fall 2012

## Audience:

This course is an advanced elective for mathematics majors who are already on-track to complete all of the major requirements. The course would also be appropriate for physics majors with sufficient mathematical background who are interested in the mathematical underpinnings of Einstein's gravitational theory of general relativity.

## Prerequisites:

MultiCalc-I and Linear Algebra are required. Differential Equations is recommended, but not required. Success in any one of Abstract Algebra, Real Analysis, or Complex Analysis is a good indication of the desired level of sophistication.

In particular, students should be comfortable taking partial derivatives and computing the rank of a matrix. Students should be familiar with the ideas of the dot product and a basis of a vector space. Students should also be able to use matrix conjugation to change the basis of a linear map, like
B = G-1AG.

## Content:

This is a first course in differential geometry at the advanced undergraduate level.

Differential geometry is a collection of mathematical methods and ideas that are used to describe the structure and symmetries of curved spaces. The subject originated with Carl Friedrich Gauss in the middle of the 19th Century. Gauss' work culminated with his Theorema Egregium (most excellent theorem), which shows that certain aspects of geometry are intrinsic to a space and do not rely on any external information.

Gauss' student, Bernhard Riemann, developed the subject significantly during the middle of the 19th century, and the most significant branch of differential geometry became called ``Riemannian geometry.'' Over several decades, Riemannian geometry was developed mathematically (by the likes of Ricci and Levi-Cevita) until it became a comprehensive subject at the beginning of the 20th century. When Albert Einstein needed a language to describe his theory of light and gravitation (general relativity), Riemannian geometry was ready for the task.

Simultaneously, another branch of differential geometry was championed by great mathematicians like Sophus Lie, Elie Cartan, and Hermann Weyl. This branch used techniques from differential geometry to study the symmetries that a space might have. By a seemingly miraculous coincidence, the theorems proven by mathematicians looked exactly like the results that physicists (notably, Paul Dirac) had just obtained by studying the orbits and quantum spin states of electrons.

Since then, differential geometry has been a major tool in all aspects of theoretical physics, from the very big (gravitation and cosmology) to the very small (quantum mechanics and particle physics) to the very weird (string theory?).

It also continues to be a major source of theoretical work in pure mathematics. Only a few years ago, a proof of the famous Poincare conjecture in topology was provided by Grigori Perelman who (very creatively!) used tools originally introduced by Ricci a century earlier.

## Outline of topics:

• The ``level-set'' description of hypersurfaces.
• The parametric description of curves and surfaces.
• The inverse and implicit function theorems as a way to move between these
• velocity vectors and the exponential map
• the tangent space and the true chain rule
• the idea of a metric as a way to measure length on a surface
• the invariance of arc-length to parametrization
• geodesics
• the Gauss map of a surface, and the shape operator of a surface
• mean curvature and scalar/Gauss curvature, and their relation to isometry (Theorema Egregium)
• remarks on Einstein's field equations and the role of curvature and geodesics to describe the shape of the universe
• (if there's time and interest) what is a Lie group?