Notes on Projective Geometry by B. Csikós

Projective Geometry

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CONTENTS

Part 1. Introduction.

The invention of perspective drawing.
The real projective space, points at infinity.
Topological structure of the real projective straight line and plane.

Part 2. Linear spaces and the associated projective spaces

Groups, rings, division rings and fields.
Vector spaces and their subspaces.
Basis, coordinates, dimension.
The projective space associated to a linear space.
Projective coordinate systems.
The theorems of Desargues and Pappus.

Part 3. Examples

Projective spaces over finite fields.
Complex projective spaces and the Hopf fibration.
Digression: Stereographic projection and inversion.
The stereographic image of the Hopf fibration.
Quaternions.

Part 4. The axiomatic treatment of projective spaces

The incidence axioms of an n-dimensional projective space.
The duality principle, the dual space.
Desargues' theorem and the incidence axioms.
Collineations.

Part 5. Desarguesian projective spaces

Construction of the division ring F.
Construction of a collineation between P and FP².

Part 6. The Fundamental Theorem of projective geometry

The Projective General Linear group.
Collineations induced by automorphisms of F.
The Fundamental Theorem of projective geometry.

Part 7. Cross-ratio preserving transformations between lines

Cross-ratio.
Characterizations of cross-ratio preserving transformations between straight lines.
Cross-ratio preserving transformations between coplanar lines.
Cross-ratio preserving transformations of a line, involutions.