28377 - Geometry 3

Learning outcomes

At the end of the course the student has a good knowledge of space curves and surfaces, and of the basis of complex analysis, with particular regard to the geometric viewpoint.

Course contents

First part:

The group of isometies of the Eudlidean space. Parametrized curves, reparametrizations, oriented parametrized curves, regular curves. Arch length for a regular curve. Curvature of a plane curve. Curvature and torsion for space curves. The local theory of curves parametrized by arc length, Frenet formulas; the local canonical form of a space curve, the fundamental theorem for space curves.

Regular surfaces, change of parameters, differential functions on surfaces. The tangent plane of a surface at a point. Orientability and the Gauss map of an orientable surface. Quadric surfaces, ruled surfaces and surfaces of revolution. The first fundamental form of a surface, length of curves on a surface, areas. Isometries, conformal mappings, equiareal maps of surfaces, a theorem of Archimedes. The second fundamental form of a surface, the Gauss and Weingarten maps, principal curvatures and principal directions, Gaussian and mean curvatures of a surface. Normal curvature and geodesic curvatures of a curve on a surface, the theorems of Meusnier and Euler. Gauss’ Theorema Egregium and the equations of compatibility. Abstract surfaces. Covariant derivative, parallel transport and geodesics, parallel transport equations and geodesic equations, Geodesics on surfaces of revolution, the theorem of Clairaut. Geodesics on the Poincare’s upper half-plane. Areas of geodesic triangles on the sphere and on the upper half-plane. Hints to spherical anf hyperbolic geometry. The Gauss-Bonnet theorem.

Second part:

Formal series. Power series. The exponential function. The argument of a complex number. Logarithms. Real and complex analytic functions. Zeros of an analytic function. Meromorphic functions.

Holomorphic functions, Cauchy-Riemann equations. Integration of a form along a path. Closed differential forms. Primitive of a closed form along a path. Integration of a closed form along homotopic paths. The Index of a point with respect to a closed path.

Cauchy's Theorem. Cauchy's Integral Formula. A holomorphic function is analytic. Morera's Theorem, Cauchy's Inequalities, Liouville's Theorem, the Fundamental Theorem of Algebra. The Maximum Principle.

Laurent series. Removable Singularities, poles and essential singularities. The Residue Theorem. Examples of the use of the Residue Theorem in the computations of real integrals.

 

Readings/Bibliography

Andrew Pressley: Elementary Differential Geometry, 2nd edition, Springer 2012

Henri Cartan: Théorie élémentaire des fonctions analytiques d'une o plusieurs variables complexes. (available in English, too)

Theodore Gamelin: Complex Analysis. Springer UTM

Teaching methods

Classroom lessons (with exercises)

Assessment methods

Exam consisting of an oral test.

The assessment is about the following aims:

-to be able to explain some part of the course, showing to be able to understand its fundamental concepts and the methods of deduction:

- to be able to solve exercises about the subject of the course.

Teaching tools

Files with exercises will be posted on the teacher's website

Office hours

See the website of Monica Idà

See the website of Mirella Manaresi