Projective Geometry on Manifolds
نویسنده
چکیده
Introduction 3 1. Affine geometry 4 1.1. Affine spaces 5 1.1.1. Euclidean geometry and its isometries 5 1.1.2. Affine spaces 7 1.1.3. Affine transformations 8 1.1.4. Tangent spaces 9 1.1.5. Acceleration and geodesics 10 1.1.6. Connections 11 1.2. The hierarchy of structures 11 1.3. Affine vector fields 12 1.4. Affine subspaces 13 1.5. Volume in affine geometry 14 1.6. Centers of gravity 14 1.7. Affine manifolds 16 1.7.1. Matrices and vector fields 16 2. Projective geometry 17 2.1. Ideal points 17 2.2. Homogeneous coordinates 18 2.2.1. The basic dictionary 21 2.2.2. Asymptotics of projective transformations 22 2.3. Affine patches 25 2.4. Projective reflections 26 2.5. Fundamental theorem of projective geometry 29 3. Duality, non-Euclidean geometry and projective metrics 30 3.1. Duality 30 3.2. Correlations and polarities 31 3.3. Intrinsic metrics 34 3.3.1. Convex cones 34 3.3.2. The Hilbert metric on convex domains 35 3.3.3. The Kobayashi metric 36
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