Math 565: Differential Geometry

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Catalog Information

Title

Differential Geometry

Credit Hours

3

Prerequisite

Math 342 or equivlaent.

Description

A rigorous treatment of the theory of differential geometry.

Desired Learning Outcomes

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include differential topology, Riemannian metrics, geodesics, curvature, and integration on manifolds.

Prerequisites

Students should have taken Math 342 prior to taking this course. Math 342 provides a rigorous background of analysis that is needed to understand many of the proofs in this course.

Minimal learning outcomes

Students should achieve mastery of the topics below. This means that they should know all relevant definitions, full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove theorems in analogy to proofs given by the instructor.

  1. Differential topology
    • Differentiable manifolds and smooth maps
    • Tangent space, tangent bundle, derivative of a smooth map
    • Immersions, submersions, and embeddings
    • Orientation
    • Vector fields, brackets
  2. Riemannian metrics
    • Definition of Riemannian metrics
    • Affine connections
    • Riemannian connections
  3. Geodesics
    • Definition of geodesics
    • Geodesic flow
    • Minimizing properties of geodesics
    • Exponential map
    • Convex neighborhoods
  4. Curvature
    • Definitions of curvature, curvature tensor
    • Second fundamental form
    • Sectional and Ricci curvature
    • Jacobi fields
  5. Integration on manifolds
    • Tensor and vector bundles
    • Exterior algebra
    • Differential forms and exterior derivative
    • Stokes Theorem






Textbooks

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

Additional topics

These are at the instructor's discretion as time allows examples are: Hopf-Rinow theorem, spaces of constant curvature, De Rham cohomology, fixed points and intersection numbers, and Morse theory.

Recommended Texts

Riemannian geometry, by Manfredo P. Do Carmo; Differential Geometry and Topology, with a view to dynamical systems, by Keith Burns and Marian Gidea

Courses for which this course is prerequisite

None.