Table of Contents

Geometry
• Introduction
• Euclidean geometry
  • EUCLID'S WORK
    • Euclid's definitions and axioms.
    • Euclid's common notions.
    • Euclid's Book I: congruence of triangles.
    • Euclid's Book I: other theorems.
    • Euclid's Book I: results independent of the parallel postulate.
    • Euclid's Books I and II: equivalence of parallelograms and dissection of rectangles.
    • Euclid's Book V: Eudoxus' treatment of incommensurables.
    • Euclid's Books III, IV, VI, XI, XII, XIII.
  • GEOMETRY AS AN ABSTRACT DOCTRINE
    • Hilbert's division of axioms.
    • Axioms of incidence.
    • Order and betweenness.
      • Axioms of order.
      • Relation of betweenness.
      • Axioms of betweenness.
      • Line defined in terms of order.
    • Results based on order and betweenness.
      • Definition of a plane.
      • Theorems concerning polygons.
      • Dissection of geometric figures.
      • Certain difficult theorems.
    • Congruence.
      • Axioms of congruence.
      • Elementary constructions.
      • Theorems that can be proved.
      • Euclid's intersecting circles.
    • A weakening of the parallel axiom.
    • Continuity.
    • Similar triangles.
    • The algebra of intervals.
    • A sample of theorems.
      • Conclusions not based on the parallel axiom.
      • Conclusions dependent upon the parallel axiom.
  • THE MEASURE OF POLYGONS AND POLYHEDRA
    • Questions following from Euclid.
    • Extension from parallelograms to polygons.
    • Equivalence of measure.
  • TRANSFORMATION GEOMETRY
    • Translation and reflection in the plane.
    • Homotheties.
    • Similitudes.
    • An algebra of points.
    • A coordinate geometry.
    • Reflections in a plane.
  • CONSTRUCTIONS
    • Constructions regarded as an existence theorem.
    • Gauge constructions.
    • Construction of radical axis of two nonconcentric circles.
    • Linear constructions.
    • Ruler and compass constructions.
    • Criterion for gauge construction.
    • Constructions with compasses only.
    • To find a circle touching three given circles.
  • GEOMETRY OF MORE THAN THREE DIMENSIONS
  • CONVEXITY
    • Convexity in Euclidean geometry.
    • Basic concepts of convexity in the development of Euclidean point set topology.
    • Convexity in Minkowskian geometry as compared to that in Euclidean geometry.
    • Geometric inequalities and convexity.
    • Construction.
• Analytic geometry
  • PLANE ANALYTIC GEOMETRY
    • Cartesian coordinates.
    • Straight lines.
      • Intersection of two lines.
      • Parametric equations.
      • Distances.
    • The circle.
    • Conic sections.
      • The ellipse.
      • The hyperbola.
      • The parabola.
    • Tangents and normals to curves.
  • PROJECTIVE AND SOLID ANALYTIC GEOMETRY
    • Analytic projective geometry.
    • Solid analytic geometry.
  • SPECIAL CURVES
    • Plane algebraic curves.
    • Plane transcendental curves.
    • General classes of curves.
    • Curves of double curvature.
• Algebraic geometry
  • BASIC CONCEPTS
    • Definition of a field.
    • Theory of polynomials.
  • PROJECTIVE AND ABSTRACT VARIETIES
    • Projective variety.
    • Abstract variety.
  • SHEAVES AND SCHEMA
    • Sheaves.
    • Schema.
  • APPLICATIONS TO ALGEBRAIC GEOMETRY
• Non-Euclidean geometry
  • HISTORY OF HYPERBOLIC GEOMETRY
  • HISTORY OF ELLIPTIC GEOMETRY
  • THE PROJECTIVE AND INVERSIVE APPROACHES
    • Projective models.
    • Inversive models.
    • The angle of parallelism.
    • Comparison of the inversive and projective models.
    • Circles and spheres.
    • Coordinates in spherical and elliptic space.
    • Coordinates in the hyperbolic plane.
  • HYPERBOLIC TRIGONOMETRY
  • TRANSFORMATIONS
• Projective geometry
  • PROJECTION
    • Perspective projection.
    • Points at infinity.
    • Projective theorems.
      • Desargues's theorem.
      • Pappus' (Pascal's) theorem.
    • Duality.
    • Projectively generated loci.
  • HOMOGENEOUS COORDINATES
    • Coordinates and transformations.
    • Geometry of one dimension.
    • The fundamental theorem.
    • Harmonic pairs.
  • COMPLEX GEOMETRY
    • Projective mappings.
    • Group properties.
  • ABSTRACT GEOMETRIES
• Differential geometry
  • MANIFOLDS AND TENSOR BUNDLES
  • OPERATIONS ON TENSOR FIELDS: CONNECTIONS
  • DE RHAM AND HODGE THEOREMS
  • GAUSS-BONNET FORMULA AND CHARACTERISTIC CLASSES
  • ELLIPTIC OPERATORS
  • ISOMETRIC IMBEDDING, SUBMANIFOLDS, AND MODERN DEVELOPMENTS IN SURFACE THEORY
    • The rigidity problem.
    • Minimal surfaces.
    • Complete surfaces.
  • COMPLEX MANIFOLDS
  • LOCAL AND GLOBAL PROBLEMS
• Riemannian geometry
  • BASIC PROPERTIES
  • APPLICATION OF RIEMANNIAN CONCEPTS
• Bibliography
  • Euclidean geometry:
  • Analytic geometry:
  • Algebraic geometry:
  • Non-Euclidean geometry:
  • Projective geometry:
  • Differential geometry:
  • Riemannian geometry: