Differential Geometry

Differential geometry is a branch of mathematics which makes use of techniques of analysis, particularly calculus, to study geometric problems. Initially, geometers primarily sought to understand the geometry of curves and surfaces in 3-dimensional Euclidean space, and many important early results in the subject are due to Gauss. Other early pioneers included Bernhard Riemann and Tullio Levi-Civita.

The primary objects of study in differential geometry are smooth and Riemannian manifolds. A typical example of such an object is a smooth surface in R^3, for example, the unit sphere. Typical questions that a differential geometer might ask about a manifold include:

• Given two points, what is the shortest path between those two points staying on the manifold? These length-minimizing paths are termed geodesics. On the sphere, the geodesics are great circles.
• Can the manifold be "flattened out"? Any piece of spaghetti can be made straight, and yet an orange peel cannot be flattened without tearing. How can we tell if it is possible to flatten a given surface? This question leads to the study of curvature.
• Is there a natural notion of distance on the manifold? In the case of a surface in R^3, the answer is yes: just define the length of a path in the surface to be the length of that path in R^3!
• How should one differentiate a function? What does it mean to take the derivative of a function on the sphere? This question leads to the development of objects called connections.

Study of differential geometry requires a firm knowledge of calculus and plane geometry.

A slight modification of differential geometry was the basis of General Relativity: the modification was replacing Riemannian manifolds for Pseudo-riemannian manifolds. In naive terms, in Riemannian manifolds the square of all distances between different points are positive numbers (so the distances can be taken to be the positive square root of their squares), while the latter uses a pseudo-metric where the square of a distance can be positive, negative or zero.