DIFFERENTIAL GEOMETRY
The study of curves and surfaces.
At the most basic level...
CURVES
Certain subsets of three dimensional space that are, in a certain sense, one-dimensional and to which the methods of differential calculus can be applied. Physically, we think of a curve as being obtained from a straight line by bending, known as curvature, and twisting, known as torsion.
SURFACES
Surfaces are subsets of three dimensional space. Differential geometry puts an emphasis on the concept of regular surfaces, obtained by taking pieces of a plane, deforming them, and arranging them in such a way that the resulting figure has no sharp points, edges, or self-intersections.
More specifically...
DIFFEOMORPHISMS
Diffeomorphisms are maps that are differentiable with a differentiable inverse. From the perspective of derivatives, two surfaces which are diffeomorphic to one another are equivalent.
GAUSSIAN CURVATURE
The most commonly considered distance on a surface is known as the Euclidean space. Gaussian curvature is a property which does not depend on this space but spec-ifically measured along the surface.
GEODESICS
A geodesic is a curve which rep-resents the shortest distance be-tween two points on a surface. It is the generalization of the notion of a straight line.
RESOURCES
Visit these sites and watch these videos to get more information about the vast subject that is differential geometry.
GENERAL DIFFERENTIAL GEOMETRY
Differential Geometry in Under 15 Minutes
Youtube video by Qilin Xue
An Intro to Differential Geometry
Article by Joseph Mellow
Differential Geometry
Youtube video by PhysicsforStudents
