Differential Geometry and Differential Manifolds

Prerequisites:

Fundamental Algebra and Analysis II

Course description:

This is a core third-year course for students in the Fundamental Mathematics major. It provides a comprehensive journey through differential geometry, beginning with the classical theory of curves and surfaces in Euclidean 3-space and advancing to the modern framework of smooth manifolds. In the first part, a central focus is the transition from extrinsic to intrinsic geometry, highlighted by Gauss's Theorema Egregium and the profound global consequence of the Gauss-Bonnet and Hilbert's Theorems. In the second part, we generalize the intrinsic viewpoint to introduce the modern framework of smooth manifolds, culminating in Stokes' Theorem and an introduction to de Rham cohomology. The entire course emphasizes an analytic perspective, rigorously connecting calculus, geometry and topology.

Teaching objectives:

By the end of this course, students will be able to:

1. Analyze curves and surfaces in Euclidean 3-space using the Frenet-Serret apparatus and the first/second fundamental forms;

2. Compute and interpret key geometric quantities such as Gaussian curvature, mean curvature, and geodesics, and explain the intrinsic nature of Gaussian curvature (Theorema Egregium);

3. State, prove, and apply the major global theorems of classical surface theory, including the Gauss-Bonnet Theorem and Hilbert's Theorem;

4. Define and work with core concepts of smooth manifold theory, including atlases, tangent spaces, vector fields, and differential forms;

5. Execute the calculus on manifolds by performing operations such as the Lie bracket, exterior derivative, and integration of forms;

6. Formulate and employ Stokes' Theorem in its general form and relate it to classical integral theorems;

7. Define de Rham cohomology groups and perform basic computations, understanding their role as a bridge between local calculus and global topology.

Detailed topics:

Part I. Classical Differential Geometry in Eculidean 3-space

1. Curves: Parametrization, curvature, torsion, Frenet-Serret frame, global theorems (Isoperimetric inequality, Four-Vertex theorem)

2. Local Surface Theory: Parametrizations, the first and second fundamental forms, Gauss map, principal curvatures, Gaussian and mean curvatures, Theorema Egregium, and intrinsic geometry

3. Global Surface Theorems: the Gauss-Bonnet Theorem and Hilbert's Theorem

Part II. Abstract Smooth Manifolds and Calculus

1. Foundations: smooth manifolds, smooth maps, tangent space, the tangent and cotangent bundles

2. Calculus on manifolds: Vector fields, integral curves and flows, Lie bracket, differential forms and the exterior algebra, exterior derivative, orientation

3. Integration Theory: Partition of unity, Integration of differential forms, Stokes' Theorem (general form and classical corollaries)

4. De Rham Cohomology: De Rham cohomology groups, computations for elementary manifolds, the Poincare lemma, cohomology perspective on Gauss-Bonnet