In Fall 2025, I will teach Differential Geometry, MATH 5433(G) A. We shall use the following course design, and use the book: Elementary Differential Geometry by Barrett O'Neill, Revised 2nd ed.
Differential geometry is the study of curves, surfaces, and higher-dimensional spaces using the tools of calculus and linear algebra. While Euclidean geometry assumes flatness, differential geometry explores spaces that can bend, twist, or curve—offering a natural language to describe non-Euclidean geometries. In this framework, the familiar concepts of distance, angle, and curvature are generalized to curved spaces, allowing us to rigorously study shapes like spheres, hyperbolic planes, and more complex manifolds.
With your background in linear algebra and multivariable calculus, you're well-prepared to grasp the key ingredients of differential geometry: vector fields, tangent spaces, and parametrized surfaces. You’ll encounter the concept of a manifold, which locally looks like Euclidean space but can have a globally curved structure. Calculus on manifolds involves generalizing derivatives (via the covariant derivative) and integrating over curved surfaces, laying the groundwork for understanding curvature more formally.
This machinery becomes essential in Einstein’s theory of general relativity, where gravity is no longer seen as a force but as the manifestation of curved spacetime. In this context, differential geometry provides the foundation for describing how mass and energy shape the geometry of the universe. Learning differential geometry is not only intellectually rich but also a gateway into modern physics, advanced geometry, and many applications in engineering and computer science.
Prerequisite: MATH 2160 Elementary Linear Algebra and MATH 2243 Calculus III.
There are four sessions, and the final exam.
20% of a session grade is from Homework Quiz, 10% is from Attendance Work, and 70% is from Session Tests.
The final exam counts for 33.3% or 50%. The percentage is determined so that it maximizes the course average.
Good work ethic and study habit are required throughout the semester.
Georgia Southern Credit Hour Policy: For each 50 min lecture, a minimum of two hours of out of class work is required.
Homework Quiz will be assigned after each class meeting, and they are collected and graded for reasonable efforts.
Homework Quiz provides students with opportunities to review concepts, practice standard problems, and challenge themselves with problem-solving skill questions which require independent and critical thinking.
There will be four 50-minute long tests, and the final exam.
Mock tests will be available before each test and the final exam.
Calculus on Euclidean Space; Frame Fields; Calculus on a Surface; Shape Operators; Geometry of Surfaces in R3; Riemannian Geometry