# MATH3531 Topology and Differential Geometry

MATH3531 is a Mathematics Level III course. See the course overview below. A higher version of this course is MATH3701.

Units of credit: 6

Prerequisites: 12 units of credit in Level 2 Math courses including MATH2011 or MATH2111.

Exclusions: MATH3690, MATH3701, MATH3531, MATH3760

Cycle of offering:T2 but subject to change in 2020

Graduate attributes: The course will enhance your research, inquiry and analytical thinking abilities.

More information: This recent course handout contains information about course objectives, assessment, course materials and the syllabus.

The Online Handbook entry contains information about the course. (The timetable is only up-to-date if the course is being offered this year.)

If you are currently enrolled in MATH3531, you can log into UNSW Moodle for this course.

#### Course Overview

This course introduces the mathematical areas of differential geometry and topology and how they are interrelated, and in particular studies various aspects of the differential geometry of surfaces. The approach to the latter taken is built around Cartan's approach, which leads more easily to modern differential geometry and also to its applications in theoretical physics.The course's major theme is how certain natural questions of "sameness" can be systematically approached and answered, and how these answers can be used.

We begin with a review extension of basic topology, multivariable calculus and linear algebra. Then we study curves and how they bend and twist in space. This will lead us to look at general ideas in the topology of curves, and the fundamental group. Then we look at one of the original themes of topology as developed by Poincare: vector fields.

Turning to differential geometry, we look at manifolds and structures on them, in particular tangent vectors and tensors. This leads to the idea of differential forms and the further topological idea of cohomology.

With these building blocks, we then consider surfaces, studying the classical fundamental forms introduced by Gauss, the various measures of curvature for surfaces and what they mean for the internal and external appearance and properties of surfaces. A closer look at the intrinsic geometry of surfaces leads to Gauss' famous "Remarkable Theorem" on curvature and provides the starting point that would lead to the fundamental uses of differential geometry in, for example, Einstein's general relativity. In relation to surfaces, we consider geodesics, the Gauss-Bonnet theorem and the Euler characteristic. These lead us to our last topic: consideration of non-Euclidean geometries, especially the hyperbolic plane.