Math 432 - Introduction to Differential Geometry
Description
Math 432 is an introduction to the theory of differential geometry. This course introduces the core concepts of differential geometry, such as curve theory, the tangent space, diffeomorphisms, differentials of smooth maps, first and second fundamental forms, orientations of surfaces, the Gauss map, minimal surfaces, and curvature, in the familiar environment of curves and surfaces in three-dimensional Euclidean space. As a result, this course prepares students for future studies in general manifold theory and its applications.
Topics covered
Topics usually covered in Math 432 include:
- Parametrized curves in Euclidean space
- Local theory of curves and arc length
- Local canonical form and the fundamental theorem of curve theory
- Regular surfaces
- Differentiable functions on regular surfaces
- The tangent plane and the differentials of smooth maps
- The first fundamental form and area
- Orientation of surfaces
- The Gauss map and its properties
- Vector fields
- Minimal surfaces
- Isometries and conformal maps
- Curvature
- The Gauss theorem and equations of compatibility
- The Gauss–Bonnet theorem
Note that the exact list of topics may vary by instructor and semester.
Comments on prerequisites
Unofficial/suggested additional prerequisites: 250, CALC4 (244, 252, Math 292).
A strong background in multivariable calculus and linear algebra is strongly recommended, but students without a strong linear algebra background may not be at a great disadvantage.
Resources
Textbook
The usual textbook for the class is Differential Geometry of Curves and Surfaces (2nd. ed.) by Manfredo do Carmo.