Non Euclidean Geometry

Code:

M351

Acronym:

M351

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2011/2012 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:AST	0	Plano de Estudos a partir de 2008	3	-	7,5	-
L:M	15	Plano de estudos a partir de 2009	3	-	7,5	-
PGMP	0	PE da PG em Matemática para Professores	1	-	7,5	-

Teaching language

Portuguese

Objectives

With greater emphasis in the metric approach (via Kleinian study of isometries), and less so in the axiomatic one, come to know the central features of the non-Euclidean (two dimensional) geometries, spherical and hyperbolic.
At a more advanced level, to relate aspects of these geometries to important topics in complex analysis (e.g. linear fractional transformations), differential geometry or topology (classification of geometric surfaces, covering spaces and transformations)

Program

1.The Euclidean Plane.
A short history of the 5th euclidean postulate, and the development of axiomatic systems for plane geometry. Distinctive features of Euclidean geometry: properties equivalente to the 5th postulate; the angle sum of a triangle, existence of similarities.
Incidence models for the spherical and hyperbolic geometries. The Poincaré’s half-plane and disc models.
Congruence and metric. The isometries of the euclidean plane: classification and the three reflexions theorem.
2.Euclidean Surfaces.
The cylinder, the twisted cylinder ( the Moebius band), the torus and the Klein bottle.
Groups of isometries in the plane and quocient surfaces: fundamental regions, discontinuous and non-discontinuos groups; examples.
Connected and complete euclidean surfaces: covering by the plane (Hopf theorem), covering isometries and the Killing-Hopf theorem.
3. Spherical Geometry
The sphere S^2 in euclidean space: geodesic metric and the three reflexions theorem.
The subgoup of rotations: Isom+(S^2) and SO(3).
Stereographic projection, inversion and comp+lex coordinates on the sphere. Geometric properties of inversion. Reflexions and rotations as complex functions.
The antipodal map and the elliptic plane. Gruops, spheres and the projectiv spaces: SO(3) and P_3(R); Quaternions and S^3.
The area of a triangle: Harriot’s theorem. Tesselations of the sphere and the regular polyedra.
4.The Hyperbolic Plane.
Negative curvature and Poincaré’s half-plane model. The pseudosphere: curvature, parametrization in the x-y plane and the associated metric in H^2.
The half-plane model and the conformal disc model (Poincaré’s).
Isometries of H^2. Basic properties of distance. Characterization of geodesic lines and reflections. The three reflections theorem. Isometries as complex functions (Poincaré’s theorem). Description and classification of isometries in H^2.
The area of a hyperbolic triangle.
The projectiv disc model. The hyperbolic space.
5.Hyperbolic surfaces
Rotations at infinity of the hyperbolic plane and cusps.: the completion of the pseudosphere; the punctured sphere; dens lines in the punctured sphere.
Construction of hyperbolic surfaces and polygons in H^2. Complet and compact hyperbolic surfaces. Completeness of the geometric compact surfaces.
The geometric classification of surfaces. Paths and geodesics in geometric surfaces; geodesic loops. Free homotopy and classification of geodesic loops.

Mandatory literature

000050577. ISBN: 0-387-97743-0

Complementary Bibliography

000052997. ISBN: 0-19-853447-7
000081896. ISBN: 0-7167-2446-4

Teaching methods and learning activities

Theoretical and practical classes. The students are supposed to participate in an active way in the theoretical classes, including lecturing on parts of the syllabus, or assigned projects, and in the practical classes by solving problems and exercises.

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	67,50
	Total:	-	0,00

Eligibility for exams

To be elegible for continuous (on-course) assessment, without final exam, students must attend 80% of theoretical classes and 80% of practical classes.

Calculation formula of final grade

In continuous assessment, there are two alternative options to attain the final grade; in the first alternative, the presentation in theoretical classes of parts of the syllabus contributes to 40%, the remaining 60% contributed through the resolution of test exercises an problems, either in the practical classes or as homework; in the second alternative, each of these percentages lower 10% and the remaining 20% are contributed by a project.
When students choose to offer the final exam, the final grade is the one obtained in that exam.

Examinations or Special Assignments

An optional Project contributing to 20% of the final grade in the continuous assessment mode.

Special assessment (TE, DA, ...)

By a single exam, according to the rules of Faculdade de Ciências

Classification improvement

By a single exam, according to the rules of Faculdade de Ciências