Mathematics BSc
Course description
2013.
Borromean Rings
Geometry 3
Hours
lect+pc		 Credits
lect+pc		 Assessment Specialization Course code
lect/pc		 Semester Status
3 + 2 3 + 3 exam +
term grade		 pure math. mm1c1ge4m
mm1c2ge4m		 4 compulsory
Strong Weak Prerequisites
Practice class
Strong:
Geometry2L-p (mm1c1ge3m)
Strong:
Algebra2L (mm1c1al2)
Lecture
Weak:
practice class
Literature
  • Marcel Berger: Geometry I. Springer, 1987.
Syllabus
  • Projective spaces. Extending Euclidean or affine space with ideal elements. Projective subspaces, dimension formula. Projective space associated with a vector space over a skew field. Representative vectors and homogeneous coordinates. Theorems of Desargues and Pappus, commutativity of the ground field. Topology of real projective plane. Complex projective spaces and the Hopf fibration.
  • Axiomatic treatment of projective spaces. Incidence axioms of n-dimensional projective space. Subspaces, dual spaces, the principle of duality. Desargues' theorem and collineations. Classification of Desarguesian projective planes, and of at least 3-dimensional projective spaces.
  • Collineations. Projective transformations and maps induced by field automorphisms in classical projective spaces. The fundamental theorem of projective geometry.
  • Cross-ratio. Definition and basic properties of cross-ratio. Cross-ratio preserving maps between projective lines. The theorem of Pappus on perspective maps. The Steiner axis. Complete quadrilaterals and harmonic separation. Involutions.
  • Pencils. Projective spaces of algebraic hypersurfaces. Examples of pencils of hyperplanes, circles, spheres, conics. Desargues' theorem on pencils of quadric hypersurfaces.
  • Conics and quadrics. Analytic definition, regularity. Conjugacy and polarity with respect to a quadric, tangents, geometric constructions of polar lines. Extending the principle of duality to quadrics. Projective classification of quadric hypersurfaces. Cross-ratio on a conic. The theorems of Pascal and Brianchon. The Steiner axis, Steiner's construction of fixed points.
  • Hyperbolic geometry. Classical geometries in the light of Klein's Erlangen Programme. Minkowski spacetime, Lorentz group, Poincaré group. Hyperboloid model of hyperbolic geometry.
  • Distance, angle, and orthogonal lines. Trigonometric formulas for triangles. The Cayley-Klein model. Orthogonality. Bolyai's definition of parallels, angle and distance of parallelism. Spheres, horospheres, and equidistance hypersurfaces in the hyperboloid model. Definitions independent of the model. The conformal models of Poincaré.
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