Math 465: Intro to Differential Geometry

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Catalog Information

Title

Introduction to Differential Geometry.

(Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

Offered

F (odd years)

Prerequisite

Math 314; 341 or equivalent.

Recommended

Math 342 or equivalent.

Description

Geometry of smooth curves and surfaces. Topics include the first and second fundamental forms, the Gauss map, orientability of surfaces, Gaussian and mean curvature, geodesics, minimal surfaces and the Gauss-Bonnet Theorem.

Desired Learning Outcomes

The main purpose of this course is to provide students with an understanding of the geometry of curves and surfaces, with the focus being on the theoretical and logical foundations of differential geometry.

Prerequisites

The prerequisite of Math 314 is to ensure that students have some understanding of partial derivatives and differentiation for functions of more than one variable. 341 is required so students have a rigorous understanding of the real number system and of real-valued functions. Math 342 is recommended so students have a rigorous understanding of functions with more than one variable.

Minimal learning outcomes

Outlined below are topics that all successful Math 465 students should understand well. As evidence of that understanding, students should be able to demonstrate mastery of all relevant vocabulary, familiarity with common examples and counterexamples, knowledge of the content of the major theorems, understanding of the ideas in their proofs, and ability to make direct application of those results to related problems.

  1. Basic Properties of Curves
    • Parametrized curves
    • Regular curves
    • Arc length
  2. Regular surfaces
    • Regular surfaces as inverse images of regular values
    • Change of parameters
    • The tangent plane
    • The first fundamental form
    • Orientation of surfaces
  3. The geometry of the Gauss map
    • Definition of the Gauss map and fundamental properties
    • The Gauss map in local coordinates
    • Minimal surfaces
  4. Intrinsic geometry of surfaces
    • Isometries and conformal maps
    • Geodesics and parallel transport
    • The Gauss-Bonnet Theorem and applications
    • The exponential map

Textbooks

Possible textbooks for this course include (but are not limited to):

  • Manfredo P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall, 1976.

Additional topics

Among other topics instructors may want to cover Jacobi fields and conjugate points, covering spaces, and the Hopf-Rinow Theorem.

Courses for which this course is prerequisite

None