Geometry and topology Miles Reid, Balazs Szendroi.
Reid, Miles| Call Number | 516.2 R27G |
| Author | Reid, Miles |
| Title | Geometry and topology Miles Reid, Balazs Szendroi. |
| Publication | Cambridge, UK ; New York : Cambridge University Press, 2005. |
| Physical Description | xviii, 196 p. : ill. ; 26 cm. |
| Contents | List of figures -- Preface -- Euclidean geometry -- metric on R n -- Lines and collinearity in R n -- Euclidean space E n-- Digression : shortest distance -- Angles -- Motions -- Motions and collinearity -- motion is affine linear on lines -- Motions are affine transformations -- Euclidean motions and orthogonal transformations -- Normal form of an orthogonal matrix -- 2 x 2 rotation and reflection matrixes -- general case -- Euclidean frames and motions -- Frames and motions Ep2s -- Every motion of Ep2s is a translation, rotation, reflection or glide -- Classification of motions of Ep3s -- Sample theorems of Euclidean geometry -- Pons asinorum -- angle sum of triangles -- Parallel lines and similar triangles -- Four centres of a triangle -- Feuerbach 9-point circle -- Composing maps -- Composition is the basic operation -- Composition of affine linear maps x [arrow] Ax + b -- Composition of two reflections of Ep2s -- Composition of maps is associative -- Decomposing motions -- Reflections generate all motions -- alternative proof of Theorem 1.14 -- Preview of transformation groups -- Exercises -- Spherical and hyperbolic non-Euclidean geometry -- Basic definitions of spherical geometry -- Spherical triangles and trig -- spherical triangle inequality -- Spherical motions -- Properties of Sp2s like Ep2s -- Properties of Sp2s unlike Ep2s -- Preview of hyperbolic geometry -- Hyperbolic space -- Hyperbolic distance -- Hyperbolic triangles and trig-- Hyperbolic motions -- Incidence of two lines in Hp2s -- hyperbolic plane is non-Euclidean -- Angular defect -- first proof -- explicit integral -- Proof by subdivision -- alternative sketch proof -- Exercises -- Affine geometry-- Motivation for affine space -- Basic properties of affine space -- geometry of affine linear subspaces -- Dimension of intersection -- Affine transformations -- Affine frames and affine transformations -- centroid -- Exercises -- Projective geometry -- Motivation for projective geometry -- Inhomogeneous to homogeneous -- Perspective -- Asymptotes -- Compactification -- Definition of projective space -- Projective linear subspaces -- Dimension of intersection -- Projective linear transformations and projective frames of reference -- Projective linear maps of Pp1s and the cross-ratio -- Perspectivities -- Affine space A n as a subset of projective space P n -- Desargues' theorem -- Pappus' theorem -- Principle of duality -- Axiomatic projective geometry -- Exercises -- Geometry and group theory -- Transformations form a group -- Transformation groups -- Klein's Erlangen program -- Conjugacy in transformation groups -- Applications of conjugacy -- Normal forms -- Finding generators -- algebraic structure of transformation groups -- Discrete reflection groups -- Exercises -- Topology -- Definition of a topological space -- Motivation from metric spaces -- Continuous maps and homeomorphisms -- Definition of a continuous map -- Definition of a homeomorphism -- Homeomorphisms and the Erlangen program -- homeomorphism problem -- Topological properties -- Connected space -- Compact space -- Continuous image of a compact space is compact -- application of topological properties -- Subspace and quotient topology-- Standard examples of glueing -- Topology of P n R -- Nonmetric quotient topologies -- Basis for a topology -- Product topology -- Hausdorff property -- Compact versus closed -- Closed maps -- criterion for homeomorphism -- Loops and the winding number -- Paths, loops and families -- winding number -- Winding number is constant in a family -- Applications of the winding number -- Quaternions, rotations and the geometry of transformation groups -- Topology on groups -- Dimension counting -- Compact and noncompact groups -- Components -- Quaternions, rotations and the geometry of SO(n) -- Quaterions -- Quaternions and rotations -- Spheres and special orthogonal groups -- group SU(2) -- electron spin in quantum mechanics -- story of the electron spin -- Measuring spin : the Stern-Gerlach device -- spin operator -- Rotate the device -- solution -- Preview of Lie groups -- Exercises -- Concluding remarks -- On the history of geometry -- Greek geometry and rigour -- parallel postulate -- Coordinates versus axioms -- Group theory -- Abstract groups versus transformation groups -- Homogeneous and principal homogeneous spaces -- Erlangen program revisited -- Affine space as a torsor -- Geometry in physcis -- 9.3.1. The Galilean group and Newtonian dynamics -- Poincare group and special relativity -- Wigner's classification : elementary particles -- Standard Model and beyond -- Other connections -- famous trichotomy -- curvature trichotomy in geometry -- On the shape and fate of the universe -- snack bar at the end of the universe -- Appendix A : Metrics -- Exercises -- Appendix B : Linear algebra -- Bilinear form and quadratic form -- Euclid and Lorentz -- Complements and bases -- Symmetries -- Orthogonal and Lorentz matrixes -- Hermitian forms and unitary matrixes -- Exercises -- References -- Index. |
| Added Author | Szendroi, Balazs. |
| Subject | GEOMETRY. TOPOLOGY. Geometrie. Topologie. |
| Multimedia |
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$a xviii, 196 p. : $b ill. ; $c 26 cm.
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$a Includes bibliographical references (p. 190-191) and index.
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$t List of figures -- $t Preface -- $g 1. $t Euclidean geometry -- $g 1.1. The $t metric on R n -- $g 1.2. $t Lines and collinearity in R n -- $g 1.3. $t Euclidean space E n-- $g 1.4. $t Digression : shortest distance -- $g 1.5. $t Angles -- $g 1.6. $t Motions -- $g 1.7. $t Motions and collinearity -- $g 1.8. A $t motion is affine linear on lines -- $g 1.9. $t Motions are affine transformations -- $g 1.10. $t Euclidean motions and orthogonal transformations -- $g 1.11. $t Normal form of an orthogonal matrix -- $g 1.11.1 The $t 2 x 2 rotation and reflection matrixes -- $g 1.11.2 The $t general case -- $g 1.12. $t Euclidean frames and motions -- $g 1.13. $t Frames and motions Ep2s -- $g 1.14. $t Every motion of Ep2s is a translation, rotation, reflection or glide -- $g 1.15. $t Classification of motions of Ep3s -- $g 1.16. $t Sample theorems of Euclidean geometry -- $g 1.16.1. $t Pons asinorum -- $g 1.16.2. The $t angle sum of triangles -- $g 1.16.3 $t Parallel lines and similar triangles -- $g 1.16.4. $t Four centres of a triangle -- $g 1.16.5. The $t Feuerbach 9-point circle -- $g Exercises -- $g 2. $t Composing maps -- $g 2.1. $t Composition is the basic operation -- $g 2.2. $t Composition of affine linear maps x [arrow] Ax + b -- $g 2.3. $t Composition of two reflections of Ep2s -- $g 2.4. $t Composition of maps is associative -- $g 2.5. $t Decomposing motions -- $g 2.6. $t Reflections generate all motions -- $g 2.7. An $t alternative proof of Theorem 1.14 -- $g 2.8. $t Preview of transformation groups -- $t Exercises --
505
0
0
$g 3. $t Spherical and hyperbolic non-Euclidean geometry -- $g 3.1. $t Basic definitions of spherical geometry -- $g 3.2. $t Spherical triangles and trig -- $g 3.3. The $t spherical triangle inequality -- $g 3.4. $t Spherical motions -- $g 3.5. $t Properties of Sp2s like Ep2s -- $g 3.6. $t Properties of Sp2s unlike Ep2s -- $g 3.7. $t Preview of hyperbolic geometry -- $g 3.8. $t Hyperbolic space -- $g 3.9. $t Hyperbolic distance -- $g 3.10. $t Hyperbolic triangles and trig-- $g 3.11. $t Hyperbolic motions -- $g 3.12. $t Incidence of two lines in Hp2s -- $g 3.13. The $t hyperbolic plane is non-Euclidean -- $g 3.14. $t Angular defect -- $g 3.14.1. The $t first proof -- $g 3.14.2. An $t explicit integral -- $g 3.14.3. $t Proof by subdivision -- $g 3.14.4. An $t alternative sketch proof -- $t Exercises -- $g 4. $t Affine geometry-- $g 4.1. $t Motivation for affine space -- $g 4.2. $t Basic properties of affine space -- $g 4.3. The $t geometry of affine linear subspaces -- $g 4.4. $t Dimension of intersection -- $g 4.5. $t Affine transformations -- $g 4.6. $t Affine frames and affine transformations -- $g 4.7. The $t centroid -- $t Exercises -- $g 5. $t Projective geometry -- $g 5.1. $t Motivation for projective geometry -- $g 5.1.1. $t Inhomogeneous to homogeneous -- $g 5.1.2. $t Perspective -- $g 5.1.3. $t Asymptotes -- $g 5.1.4. $t Compactification -- $g 5.2. $t Definition of projective space -- $g 5.3. $t Projective linear subspaces -- $g 5.4. $t Dimension of intersection -- $g 5.5. $t Projective linear transformations and projective frames of reference -- $g 5.6. $t Projective linear maps of Pp1s and the cross-ratio --
505
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$g 5.7. $t Perspectivities -- $g 5.8. $t Affine space A n as a subset of projective space P n -- $g 5.9. $t Desargues' theorem -- $g 5.10. $t Pappus' theorem -- $g 5.11. $t Principle of duality -- $g 5.12. $t Axiomatic projective geometry -- $t Exercises -- $g 6. $t Geometry and group theory -- $g 6.1. $t Transformations form a group -- $g 6.2. $t Transformation groups -- $g 6.3. $t Klein's Erlangen program -- $g 6.4. $t Conjugacy in transformation groups -- $g 6.5. $t Applications of conjugacy -- $g 6.5.1. $t Normal forms -- $g 6.5.2. $t Finding generators -- $g 6.5.3. The $t algebraic structure of transformation groups -- $g 6.6. $t Discrete reflection groups -- $t Exercises -- $g 7. $t Topology -- $g 7.1. $t Definition of a topological space -- $g 7.2. $t Motivation from metric spaces -- $g 7.3. $t Continuous maps and homeomorphisms -- $g 7.3.1. $t Definition of a continuous map -- $g 7.3.2. $t Definition of a homeomorphism -- $g 7.3.3. $t Homeomorphisms and the Erlangen program -- $g 7.3.4. The $t homeomorphism problem -- $g 7.4. $t Topological properties -- $g 7.4.1. $t Connected space -- $g 7.4.2. $t Compact space -- $g 7.4.3. $t Continuous image of a compact space is compact -- $g 7.4.4. An $t application of topological properties -- $g 7.5. $t Subspace and quotient topology-- $g 7.6. $t Standard examples of glueing --
505
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$g 7.7. $t Topology of P n R -- $g 7.8. $t Nonmetric quotient topologies -- $g 7.9. $t Basis for a topology -- $g 7.10. $t Product topology -- $g 7.11. The $t Hausdorff property -- $g 7.12. $t Compact versus closed -- $g 7.13. $t Closed maps -- $g 7.14. A $t criterion for homeomorphism -- $g 7.15. $t Loops and the winding number -- $g 7.15.1. $t Paths, loops and families -- $g 7.15.2. The $t winding number -- $g 7.15.3. $t Winding number is constant in a family -- $g 7.15.4. $t Applications of the winding number -- $g Exercises -- $g 8. $t Quaternions, rotations and the geometry of transformation groups -- $g 8.1. $t Topology on groups -- $g 8.2. $t Dimension counting -- $g 8.3. $t Compact and noncompact groups -- $g 8.4. $t Components -- $g 8.5. $t Quaternions, rotations and the geometry of SO(n) -- $g 8.5.1. $t Quaterions -- $g 8.5.2. $t Quaternions and rotations -- $g 8.5.3. $t Spheres and special orthogonal groups -- $g 8.6. The $t group SU(2) -- $g 8.7. The $t electron spin in quantum mechanics -- $g 8.7.1. The $t story of the electron spin -- $g 8.7.2. $t Measuring spin : the Stern-Gerlach device -- $g 8.7.3. The $t spin operator -- $g 8.7.4. $t Rotate the device -- $g 8.7.5. The $t solution -- $g 8.8. $t Preview of Lie groups -- $t Exercises --
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$g 9. $t Concluding remarks -- $g 9.1. $t On the history of geometry -- $g 9.1.1. $t Greek geometry and rigour -- $g 9.1.2. The $t parallel postulate -- $g 9.1.3. $t Coordinates versus axioms -- $g 9.2. $t Group theory -- $g 9.2.1. $t Abstract groups versus transformation groups -- $g 9.2.2. $t Homogeneous and principal homogeneous spaces -- $g 9.2.3. The $t Erlangen program revisited -- $g 9.2.4. $t Affine space as a torsor -- $g 9.3. $t Geometry in physcis -- $t 9.3.1. The $t Galilean group and Newtonian dynamics -- $g 9.3.2. The $t Poincare group and special relativity -- $g 9.3.3. $t Wigner's classification : elementary particles -- $g 9.3.4. The $t Standard Model and beyond -- $g 9.3.5. $t Other connections -- $g 9.4. The $t famous trichotomy -- $g 9.4.1. The $t curvature trichotomy in geometry -- $g 9.4.2. $t On the shape and fate of the universe -- $g 9.4.3. The $t snack bar at the end of the universe -- $t Appendix A : Metrics -- $t Exercises -- $t Appendix B : Linear algebra -- $g B.1. $t Bilinear form and quadratic form -- $g B.2. $t Euclid and Lorentz -- $g B.3. $t Complements and bases -- $g B.4. $t Symmetries -- $g B.5. $t Orthogonal and Lorentz matrixes -- $g B.6. $t Hermitian forms and unitary matrixes -- $t Exercises -- $t References -- $t Index.
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$a GEOMETRY.
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| Contents | List of figures -- Preface -- Euclidean geometry -- metric on R n -- Lines and collinearity in R n -- Euclidean space E n-- Digression : shortest distance -- Angles -- Motions -- Motions and collinearity -- motion is affine linear on lines -- Motions are affine transformations -- Euclidean motions and orthogonal transformations -- Normal form of an orthogonal matrix -- 2 x 2 rotation and reflection matrixes -- general case -- Euclidean frames and motions -- Frames and motions Ep2s -- Every motion of Ep2s is a translation, rotation, reflection or glide -- Classification of motions of Ep3s -- Sample theorems of Euclidean geometry -- Pons asinorum -- angle sum of triangles -- Parallel lines and similar triangles -- Four centres of a triangle -- Feuerbach 9-point circle -- Composing maps -- Composition is the basic operation -- Composition of affine linear maps x [arrow] Ax + b -- Composition of two reflections of Ep2s -- Composition of maps is associative -- Decomposing motions -- Reflections generate all motions -- alternative proof of Theorem 1.14 -- Preview of transformation groups -- Exercises -- Spherical and hyperbolic non-Euclidean geometry -- Basic definitions of spherical geometry -- Spherical triangles and trig -- spherical triangle inequality -- Spherical motions -- Properties of Sp2s like Ep2s -- Properties of Sp2s unlike Ep2s -- Preview of hyperbolic geometry -- Hyperbolic space -- Hyperbolic distance -- Hyperbolic triangles and trig-- Hyperbolic motions -- Incidence of two lines in Hp2s -- hyperbolic plane is non-Euclidean -- Angular defect -- first proof -- explicit integral -- Proof by subdivision -- alternative sketch proof -- Exercises -- Affine geometry-- Motivation for affine space -- Basic properties of affine space -- geometry of affine linear subspaces -- Dimension of intersection -- Affine transformations -- Affine frames and affine transformations -- centroid -- Exercises -- Projective geometry -- Motivation for projective geometry -- Inhomogeneous to homogeneous -- Perspective -- Asymptotes -- Compactification -- Definition of projective space -- Projective linear subspaces -- Dimension of intersection -- Projective linear transformations and projective frames of reference -- Projective linear maps of Pp1s and the cross-ratio -- Perspectivities -- Affine space A n as a subset of projective space P n -- Desargues' theorem -- Pappus' theorem -- Principle of duality -- Axiomatic projective geometry -- Exercises -- Geometry and group theory -- Transformations form a group -- Transformation groups -- Klein's Erlangen program -- Conjugacy in transformation groups -- Applications of conjugacy -- Normal forms -- Finding generators -- algebraic structure of transformation groups -- Discrete reflection groups -- Exercises -- Topology -- Definition of a topological space -- Motivation from metric spaces -- Continuous maps and homeomorphisms -- Definition of a continuous map -- Definition of a homeomorphism -- Homeomorphisms and the Erlangen program -- homeomorphism problem -- Topological properties -- Connected space -- Compact space -- Continuous image of a compact space is compact -- application of topological properties -- Subspace and quotient topology-- Standard examples of glueing -- Topology of P n R -- Nonmetric quotient topologies -- Basis for a topology -- Product topology -- Hausdorff property -- Compact versus closed -- Closed maps -- criterion for homeomorphism -- Loops and the winding number -- Paths, loops and families -- winding number -- Winding number is constant in a family -- Applications of the winding number -- Quaternions, rotations and the geometry of transformation groups -- Topology on groups -- Dimension counting -- Compact and noncompact groups -- Components -- Quaternions, rotations and the geometry of SO(n) -- Quaterions -- Quaternions and rotations -- Spheres and special orthogonal groups -- group SU(2) -- electron spin in quantum mechanics -- story of the electron spin -- Measuring spin : the Stern-Gerlach device -- spin operator -- Rotate the device -- solution -- Preview of Lie groups -- Exercises -- Concluding remarks -- On the history of geometry -- Greek geometry and rigour -- parallel postulate -- Coordinates versus axioms -- Group theory -- Abstract groups versus transformation groups -- Homogeneous and principal homogeneous spaces -- Erlangen program revisited -- Affine space as a torsor -- Geometry in physcis -- 9.3.1. The Galilean group and Newtonian dynamics -- Poincare group and special relativity -- Wigner's classification : elementary particles -- Standard Model and beyond -- Other connections -- famous trichotomy -- curvature trichotomy in geometry -- On the shape and fate of the universe -- snack bar at the end of the universe -- Appendix A : Metrics -- Exercises -- Appendix B : Linear algebra -- Bilinear form and quadratic form -- Euclid and Lorentz -- Complements and bases -- Symmetries -- Orthogonal and Lorentz matrixes -- Hermitian forms and unitary matrixes -- Exercises -- References -- Index. |
| Subject | GEOMETRY. TOPOLOGY. Geometrie. Topologie. |
| Multimedia |