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Department of Mathematical Sciences

# MATH3021 Differential Geometry III

Differential geometry is the study of curvature. Historically it arose from the application of the differential calculus to the study of curves and surfaces in 3-dimensional Euclidean space. Today it is an area of very active research mainly concerned with the higher-dimensional analogues of curves and surfaces which are known as n-dimensional differentiable manifolds, although there has been a great revival of interest in surfaces in recent years. Differential geometry has been strongly influenced by a wide variety of ideas from mathematics and the physical sciences. For instance, the surface formed by a soap film spanning a wire loop is an example of a minimal surface (that is, a surface whose mean curvature is zero) but the ideas and techniques involved in analysing and characterising such surfaces arose from the calculus of variations and from Riemann's attempt to understand complex analysis geometrically. The interplay of ideas from different branches of mathematics and the way in which it can be used to describe the physical world (as in the case of the theory of relativity) are just two features which make differential geometry so interesting.

In order to keep the treatment as elementary and intuitive as possible this level III course will be almost entirely devoted to the differential geometry of curves and surfaces, although most of the material readily extends to higher dimensions. The techniques used are a mixture of calculus, linear algebra, and topology, with perhaps a little material from complex analysis and differential equations. The course will follow the Woodward and Bolton book. The book by Do Carmo is also very suitable.

## Outline of Course

Aim: To provide a basic introduction to the theory of curves and surfaces, mostly in 3-dimensional Euclidean space. The essence of the module is the understanding of differential geometric ideas using a selection of carefully chosen interesting examples.

### Term 1

• Curves: Plane curves. Arc length, unit tangent and normal vectors, signed curvature, Fundamental theorem. Involutes and evolutes. Gauss map, global properties. Space curves. Serret-Frenet formulae. Fundamental theorem. Global properties.
• Surfaces in ℝn: Brief review of functions of several variables including differential (with geometric interpretation) and Inverse Function Theorem. Definition of regular surface in ℝn. Change of coordinates. Curves on a surface, tangent planes to a surface in ℝn.
• First Fundamental Form: Metric, length, angle, area. Orthogonal coordinates. Orthogonal families of curves on a surface. Some more abstract notions such as "metrics" on open subsets of ℝn can be introduced here.
• Mappings of Surfaces: Definitions, differential, expressions in terms of local coordinates. Conformal mappings and local isometrics. Examples.

### Term 2

• Geometry of the Gauss Map: Gauss map and Weingarten map for surfaces in ℝ³ , second fundamental form, normal (and geodesic) curvature, principal curvatures and directions. Gaussian curvature K and mean curvature H. Second fundamental form as second order approximation to the surface at a point. Explicit calculations of the above in local coordinates. Umbilics. Compact surface in ℝ³ has an elliptic point.
• Intrinsic Metric Properties: Christoffel symbols, Theorema Egregium of Gauss. Expression of K in terms of the first fundamental form. Examples. Intrinsic descriptions of K using arc length or area. Mention of Bonnet's Theorem. Surfaces of constant curvature.
• Geodesics: Definition and different characterisations of geodesics. Geodesics on surfaces of revolution. Geodesic curvature with description in terms of local coordinates.
• Minimal Surfaces: Definition and different characterisations of minimal surfaces. Conjugate minimal surfaces and the associated family. Weierstrass representation.
• Gauss-Bonnet Theorem: Gauss-Bonnet Theorem for a triangle. Angular defect. Relationship between curvature and geometry. Global Gauss-Bonnet Theorem. Corollaries of Gauss-Bonnet Theorem. The Euler-Poincaré-Hopf Theorem.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.