MATH 4351 - Calculus on Manifolds - University of Houston

# MATH 4351 - Calculus on Manifolds

***This is a course guideline.  Students should contact instructor for the updated information on current course syllabus, textbooks, and course content***

Prerequisites: MATH 4350

Course Description: Continuation of the study of Differential Geometry from MATH 4350. Holonomy and the Gauss-Bonnet theorem, introduction to hyperbolic geometry, surface theory with differential forms, calculus of variations and surfaces of constant mean curvature, abstract surfaces (2D Riemannian manifolds).

Textbook: Instructor's Notes. Reference book: Differential Geometry: A first course in curves and surfaces, Preliminary Version Summer 2016 by Prof. Theodore Shifrin.

Topics Covered: *This is the second part of the year-long course. We will finish the rest of the textbook, and then will cover some advanced topics.*

This year-long course will introduce the theory of the geometry of curves and surfaces in three-dimensional space using calculus techniques, exhibiting the interplay between local and global quantities.

Chapters include:

• Chapter 1: Some preparation knowledge (in this chapter, we review some knowledge needed from calculus and linear algebra. In particular you should pay attention on the concept of ”directional derivatives” which will constantly be used)
• Chapter 2: The Geometry of Curves (This chapter contains the main topics of the curves: The curvature and torsion (and their geometric meanings), the Frenet frame, the Frenet formula)
• Chapter 3: Surfaces in R3 (n this chapter, we introduce the tangent spaces, the first and second fundamental forms, and the shape operator for surfaces. The first fundamental form gives the measurement (i.e. the length of the curve and surface area). The second fundamental form and the shape operator will be used in the next chapter to define various concepts of curvatures)
• Chapter 4: Curvature (In this chapter, we introduce the concepts of various curvatures, including the normal, principal, Gauss and mean curvatures. We also study the curves on the surfaces. For curves, we have the concepts of the normal and geodesic curvatures)
• Chapter 5: Intrinsic geometry of surfaces (In this chapter, we use the ”moving frame” method to prove the Gauss’ remarkable theorem: the Gauss curvature is indeed an intrinsic quantity, i.e. the quantity can be observed by an inhabitant (for example, a very thin ant) of the surface, who can only perceive what happens along (or, say, tangential to) the surface. Other intrinsic properties, like the covariant derivatives, and the geodesics are also discussed.
• Finally, the Gauss-Bonnet theorem is proved. The Gauss-Bonnet theorem is the ”crowning result” in mathematics). Chapter 1 and 2 will be covered in Exam 1. Exam 2 will cover Chapter 3 and (part of) Chapter 4. Exam 3 will cover (part of) Chapter 4 and Chapter 5. In the second semester, depending on the interests of the audience, some advanced topics can be discussed as well.