Lectures on Geometry

✍ Scribed by Lucian Bădescu , Ettore Carletti

• Publisher: Springer Nature Switzerland
• Year: 2024
• Tongue: English
• Leaves: 493
• Series: UNITEXT 158
• Edition: 1
• Category: Library

✦ Synopsis

This is an introductory textbook on geometry (affine, Euclidean and projective) suitable for any undergraduate or first-year graduate course in mathematics and physics. In particular, several parts of the first ten chapters can be used in a course of linear algebra, affine and Euclidean geometry by students of some branches of engineering and computer science. Chapter 11 may be useful as an elementary introduction to algebraic geometry for advanced undergraduate and graduate students of mathematics. Chapters 12 and 13 may be a part of a course on non-Euclidean geometry for mathematics students. Chapter 13 may be of some interest for students of theoretical physics (Galilean and Einstein’s general relativity). It provides full proofs and includes many examples and exercises. The covered topics include vector spaces and quadratic forms, affine and projective spaces over an arbitrary field; Euclidean spaces; some synthetic affine, Euclidean and projective geometry; affine and projective hyperquadrics with coefficients in an arbitrary field of characteristic different from 2; Bézout’s theorem for curves of P^2 (K), where K is a fixed algebraically closed field of arbitrary characteristic; and Cayley-Klein geometries.

✦ Table of Contents

Preface
Contents
1 Linear Algebra
1.1 Introductory Notions
1.1.1 Matrices and Linear Systems
1.2 Vector Spaces
1.2.1 Vector Subspaces, Generators, Linearly Independent Sets, Bases and Dimension of a Vector Space
1.3 Linear Maps
1.3.1 The Dual Vector Space
1.3.2 Direct Sum of Vector Subspaces
1.3.3 Eigenvalues and Eigenvectors
1.4 Euclidean Vector Spaces
1.4.1 Orthogonal Automorphisms
1.4.2 Self-Adjoint Operators
1.5 Exercises
2 Bilinear and Quadratic Forms
2.1 Bilinear Maps
2.1.1 Diagonalization of Quadratic Forms
2.2 Cross-Product
2.3 Exercises
3 Affine Spaces
3.1 Affine Spaces
3.1.1 The Affine Ratio and Menelaus' Theorem
3.1.2 The Affine Subspaces of An(K)
3.2 Affine Morphisms
3.2.1 Dimension Theorem
3.2.2 Projections and Symmetries
3.2.3 Thales' and Ceva's Theorems
3.2.4 Real Affine Spaces and Convex Sets
3.3 Exercises
4 Euclidean Spaces
4.1 Euclidean Affine Spaces
4.1.1 Orthogonality
4.2 Orthogonal Affine Morphisms
4.2.1 Structure of Isometries
4.3 Exercises
5 Affine Hyperquadrics
5.1 Affine Hypersurfaces
5.1.1 Tangent Hyperplanes and Multiple Points
5.2 Affine Hyperquadrics
5.2.1 The Reduced Equation of a Hyperquadric of An(K)
5.2.2 Euclidean Classification of the Real Hyperquadrics
5.3 Real Conics
5.4 Real Quadrics
5.5 Exercises
6 Projective Spaces
6.1 Some Elementary Synthetic Projective Geometry
6.1.1 General Projective Spaces
6.2 The Projective Space Associated with a K-Vector Space
6.2.1 Projective Subspaces of the Standard Projective Space
6.3 Dual Projective Space and Projective Duality
6.4 Exercises
7 Desargues' Axiom
7.1 Exercises
8 General Linear Projective Automorphisms
8.1 Projective Homotheties and Translations
8.1.1 Desargues' Axiom, Pappus' Axiom and the Division Ring of the Coordinates
8.2 The Group of Projective Automorphisms of Pn(K)
8.2.1 Geometric Characterization of the Field K
8.3 Exercises
9 Affine Geometry and Projective Geometry
9.1 Affine Space Structure on the Complement of a Hyperplane of Pn(K)
9.2 Projective Closure of an Affine Subspace
9.2.1 Projective Automorphisms and AffineAutomorphisms
9.3 Cross-Ratio (or Anharmonic Ratio) of Four Collinear Points
9.3.1 Harmonic Ratio
9.3.2 Involutions of P1(K)
9.3.3 Homographies of the Complex Affine Line A1(C)
9.4 Exercises
10 Projective Hyperquadrics
10.1 Projective Hypersurfaces
10.1.1 Smooth Points and Tangent Hyperplanes
10.2 Projective Hyperquadrics
10.2.1 Reduced Equations of Projective Hyperquadrics
10.3 Polarity with Respect to a Hyperquadric
10.3.1 Affine and Euclidean Geometry in a ProjectiveSetting
10.4 Exercises
11 Bézout's Theorem for Curves of P2(K)
11.1 Proof of a Simple Case of Weak Bézout's Theorem
11.1.1 Two Applications of Bézout's Theorem
11.2 The General Bézout's Theorem and Further Applications
11.3 The Resultant of Two Polynomials
11.4 Intersection Multiplicity of Two Curves
11.5 Applications of Bézout's Theorem
11.5.1 Points of Inflection
11.5.2 Legendre Form of Cubics
11.5.3 Max Noether's Theorem
11.5.4 Conics Passing Through a Finite Number of Points
11.6 Exercises
12 Absolute Plane Geometry
12.1 Elements of Absolute Plane Geometry
12.1.1 Angles in an Absolute Plane
12.1.2 Triangles
12.2 The Poincaré Hyperbolic Plane
12.3 Exercises
13 Cayley–Klein Geometries
13.1 Euclidean Metric from a Projective Point of View
13.2 Projective Metrics on P1(R)
13.3 Projective Metrics of P2(R) and P3(R)
13.3.1 Non-degenerate Absolute
13.3.2 Non-ruled Quadric (i.e. i(F) = 2) F = x02 + x12 + x22 - x32. We Get Four Actual Spaces
13.3.3 Ruled Quadric (i.e. i(F) = 1) F = x02 + x12 - x22 - x32. We Get Three Actual Spaces On X = P3(R) V+(F)
13.3.4 Degenerate Absolute
13.4 General Absolutes
13.5 Exercises
References
Index

✦ Subjects

Affine Geometry, Euclidian Geometry, Projective Geometry, Cayley-Klein Geometry