\contentsline {chapter}{\hbox to\@tempdima {0\hfil }Introduction}{v}
\contentsline {section}{\numberline {0.1}What is geometry about?}{v}
\contentsline {section}{\numberline {0.2}Change of coordinates, motions, group theory and the \\ Erlangen program}{vi}
\contentsline {section}{\numberline {0.3}Geometry in applications}{viii}
\contentsline {section}{\numberline {0.4}About this book}{ix}
\contentsline {paragraph}{Coordinate geometry:}{ix}
\contentsline {paragraph}{Linear algebra:}{ix}
\contentsline {paragraph}{Multilinear algebra:}{ix}
\contentsline {paragraph}{Metric spaces:}{ix}
\contentsline {paragraph}{Group theory:}{ix}
\contentsline {chapter}{Table of contents}{xiii}
\contentsline {chapter}{List of figures}{xix}
\contentsline {chapter}{\numberline {1}Euclidean geometry}{1}
\contentsline {section}{\numberline {1.1}The metric on $\R ^n$}{1}
\contentsline {paragraph}{Proof}{2}
\contentsline {section}{\numberline {1.2}Lines and collinearity in $\@mathbb R^n$}{3}
\contentsline {paragraph}{Definition}{3}
\contentsline {section}{\numberline {1.3}Euclidean space $\@mathbb E^n$}{4}
\contentsline {paragraph}{Remark}{4}
\contentsline {section}{\numberline {1.4}Digression: shortest distance}{5}
\contentsline {paragraph}{Sketch Proof}{5}
\contentsline {section}{\numberline {1.5}Angles}{6}
\contentsline {section}{\numberline {1.6}Motions}{7}
\contentsline {section}{\numberline {1.7}Motions and collinearity}{8}
\contentsline {paragraph}{Proof}{8}
\contentsline {section}{\numberline {1.8}A motion is affine linear on lines}{8}
\contentsline {paragraph}{Proof}{8}
\contentsline {section}{\numberline {1.9}Motions are affine transformations}{9}
\contentsline {paragraph}{Definition}{9}
\contentsline {paragraph}{Discussion}{9}
\contentsline {paragraph}{Proof}{9}
\contentsline {paragraph}{Remark}{10}
\contentsline {section}{\numberline {1.10}Euclidean motions and orthogonal transformations}{11}
\contentsline {paragraph}{Proof}{11}
\contentsline {paragraph}{Definition}{11}
\contentsline {section}{\numberline {1.11}Normal form of an orthogonal matrix}{12}
\contentsline {subsection}{\numberline {1.11.1}The $2\times 2$ rotation and reflection matrixes}{12}
\contentsline {subsection}{\numberline {1.11.2}The general case}{13}
\contentsline {paragraph}{Discussion}{14}
\contentsline {paragraph}{Proof}{14}
\contentsline {paragraph}{Step 1}{14}
\contentsline {paragraph}{Step 2}{14}
\contentsline {paragraph}{Step 3}{15}
\contentsline {paragraph}{Step 4}{15}
\contentsline {paragraph}{Step 5. Proof of the theorem}{15}
\contentsline {section}{\numberline {1.12}Euclidean frames and motions}{15}
\contentsline {paragraph}{Definition}{15}
\contentsline {paragraph}{Remark}{16}
\contentsline {paragraph}{Proof}{16}
\contentsline {section}{\numberline {1.13}Frames and motions of\/ $\E ^2$}{16}
\contentsline {section}{\numberline {1.14}Classification of motions of\/ $\E ^2$}{17}
\contentsline {paragraph}{Proof}{17}
\contentsline {paragraph}{Step 1}{18}
\contentsline {paragraph}{Step 2}{18}
\contentsline {section}{\numberline {1.15}Classification of motions of\/ $\@mathbb E^3$}{18}
\contentsline {paragraph}{Proof}{20}
\contentsline {section}{\numberline {1.16}Sample theorems of Euclidean geometry}{21}
\contentsline {subsection}{\numberline {1.16.1}Pons asinorum}{22}
\contentsline {paragraph}{Proof}{22}
\contentsline {subsection}{\numberline {1.16.2}The angle sum of triangles}{22}
\contentsline {paragraph}{Proof}{23}
\contentsline {paragraph}{Remark}{23}
\contentsline {subsection}{\numberline {1.16.3}Parallel lines and similar triangles}{23}
\contentsline {paragraph}{Proof}{24}
\contentsline {subsection}{\numberline {1.16.4}Four centres of a triangle}{25}
\contentsline {paragraph}{Proof}{25}
\contentsline {paragraph}{Proof}{25}
\contentsline {paragraph}{Proof}{27}
\contentsline {paragraph}{Proof}{27}
\contentsline {subsection}{\numberline {1.16.5}The Feuerbach 9-point circle}{27}
\contentsline {paragraph}{Proof}{27}
\contentsline {section}{Exercises to Chapter\nobreakspace {}1}{28}
\contentsline {chapter}{\numberline {2}Composing maps}{33}
\contentsline {section}{\numberline {2.1}Composition is the basic operation}{33}
\contentsline {paragraph}{Definition}{33}
\contentsline {section}{\numberline {2.2}Composition of affine linear maps $\@mathbf x\DOTSB \mapstochar \rightarrow A\@mathbf x+\@mathbf b$}{34}
\contentsline {section}{\numberline {2.3}Composition of two reflections of $\@mathbb E^2$}{35}
\contentsline {section}{\numberline {2.4}Composition of maps is associative}{35}
\contentsline {section}{\numberline {2.5}Decomposing motions}{36}
\contentsline {paragraph}{Definition}{36}
\contentsline {paragraph}{Definition}{36}
\contentsline {paragraph}{Proof}{37}
\contentsline {section}{\numberline {2.6}Reflections generate all motions}{37}
\contentsline {paragraph}{Proof}{37}
\contentsline {section}{\numberline {2.7}An alternative proof of Theorem\nobreakspace {}1.14\hbox {}}{38}
\contentsline {paragraph}{Proof}{38}
\contentsline {section}{\numberline {2.8}Preview of transformation groups}{39}
\contentsline {section}{Exercises to Chapter\nobreakspace {}2}{40}
\contentsline {chapter}{\numberline {3}Non-Euclidean geometry}{43}
\contentsline {section}{\numberline {3.1}Basic definitions of spherical geometry}{45}
\contentsline {paragraph}{Remarks}{45}
\contentsline {section}{\numberline {3.2}Spherical triangles and trig}{46}
\contentsline {paragraph}{Proof}{47}
\contentsline {section}{\numberline {3.3}The spherical triangle inequality}{48}
\contentsline {paragraph}{Proof}{48}
\contentsline {section}{\numberline {3.4}Spherical motions}{48}
\contentsline {paragraph}{Proof}{49}
\contentsline {section}{\numberline {3.5}Properties of $S^2$ like $\@mathbb E^2$}{49}
\contentsline {section}{\numberline {3.6}Properties of $S^2$ unlike $\@mathbb E^2$}{50}
\contentsline {paragraph}{Proof}{51}
\contentsline {section}{\numberline {3.7}Preview of hyperbolic geometry}{52}
\contentsline {section}{\numberline {3.8}Hyperbolic space}{53}
\contentsline {section}{\numberline {3.9}Hyperbolic distance}{54}
\contentsline {paragraph}{Proof}{54}
\contentsline {paragraph}{Remark}{55}
\contentsline {section}{\numberline {3.10}Hyperbolic triangles and trig}{55}
\contentsline {paragraph}{Proof}{56}
\contentsline {paragraph}{Proof}{56}
\contentsline {paragraph}{Remark}{57}
\contentsline {section}{\numberline {3.11}Hyperbolic motions}{57}
\contentsline {paragraph}{Proof}{57}
\contentsline {paragraph}{Remark}{58}
\contentsline {section}{\numberline {3.12}Incidence of two lines in $\@mathcal H^2$}{59}
\contentsline {paragraph}{Definition}{60}
\contentsline {section}{\numberline {3.13}The hyperbolic plane is non-Euclidean}{60}
\contentsline {paragraph}{Remark}{62}
\contentsline {paragraph}{Proof}{62}
\contentsline {paragraph}{Discussion}{62}
\contentsline {paragraph}{Other models}{63}
\contentsline {section}{\numberline {3.14}Angular defect}{63}
\contentsline {subsection}{\numberline {3.14.1}The first proof}{63}
\contentsline {subsection}{\numberline {3.14.2}An explicit integral}{64}
\contentsline {paragraph}{Proof}{64}
\contentsline {subsection}{\numberline {3.14.3}Proof by subdivision}{66}
\contentsline {paragraph}{Proof}{66}
\contentsline {paragraph}{Proof of Theorem\nobreakspace {}3.14\hbox {}}{67}
\contentsline {subsection}{\numberline {3.14.4}An alternative sketch proof}{67}
\contentsline {paragraph}{Proof}{69}
\contentsline {paragraph}{Proof}{69}
\contentsline {section}{Exercises to Chapter\nobreakspace {}3}{69}
\contentsline {paragraph}{\rm 3\hbox {}.3\hbox {}}{76}
\contentsline {paragraph}{\rm 3\hbox {}.4\hbox {}}{76}
\contentsline {paragraph}{\rm 3\hbox {}.5\hbox {}}{76}
\contentsline {paragraph}{\rm 3\hbox {}.6\hbox {}}{76}
\contentsline {paragraph}{\rm 3\hbox {}.9\hbox {}}{76}
\contentsline {paragraph}{\rm 3\hbox {}.13\hbox {}}{77}
\contentsline {paragraph}{\rm 3\hbox {}.14\hbox {}}{77}
\contentsline {paragraph}{\rm 3\hbox {}.15\hbox {}}{77}
\contentsline {paragraph}{\rm 3\hbox {}.18\hbox {}}{77}
\contentsline {paragraph}{\rm 3\hbox {}.19\hbox {}}{77}
\contentsline {chapter}{\numberline {4}Affine geometry}{79}
\contentsline {section}{\numberline {4.1}Motivation for affine space}{79}
\contentsline {section}{\numberline {4.2}Basic properties of affine space}{81}
\contentsline {paragraph}{Remarks}{82}
\contentsline {section}{\numberline {4.3}The geometry of affine linear subspaces}{82}
\contentsline {paragraph}{Definition}{84}
\contentsline {section}{\numberline {4.4}Dimension of intersection}{85}
\contentsline {paragraph}{Proof}{85}
\contentsline {section}{\numberline {4.5}Affine transformations}{86}
\contentsline {paragraph}{Definition}{86}
\contentsline {section}{\numberline {4.6}Affine frames and affine transformations}{86}
\contentsline {paragraph}{Definition}{86}
\contentsline {section}{\numberline {4.7}The centroid}{87}
\contentsline {paragraph}{Proof}{87}
\contentsline {section}{Exercises to Chapter\nobreakspace {}4}{88}
\contentsline {chapter}{\numberline {5}Projective geometry}{91}
\contentsline {section}{\numberline {5.1}Motivation for projective geometry}{91}
\contentsline {subsection}{\numberline {5.1.1}Inhomogeneous to homogeneous}{91}
\contentsline {subsection}{\numberline {5.1.2}Perspective}{92}
\contentsline {subsection}{\numberline {5.1.3}Asymptotes}{93}
\contentsline {subsection}{\numberline {5.1.4}Compactification}{94}
\contentsline {section}{\numberline {5.2}Definition of projective space}{95}
\contentsline {section}{\numberline {5.3}Projective linear subspaces}{96}
\contentsline {paragraph}{Definition}{97}
\contentsline {section}{\numberline {5.4}Dimension of intersection}{97}
\contentsline {paragraph}{Proof}{97}
\contentsline {section}{\numberline {5.5}$\PGL (n+1)$ and projective frames of reference}{97}
\contentsline {paragraph}{Definition}{98}
\contentsline {paragraph}{Proof}{99}
\contentsline {section}{\numberline {5.6}Projective linear maps of $\PP ^1$ and the cross-ratio}{99}
\contentsline {paragraph}{Remark}{100}
\contentsline {paragraph}{Proof}{100}
\contentsline {section}{\numberline {5.7}Perspectivities}{101}
\contentsline {paragraph}{Proof}{102}
\contentsline {section}{\numberline {5.8}Affine space as a subset of projective space}{102}
\contentsline {section}{\numberline {5.9}Desargues' theorem}{103}
\contentsline {paragraph}{Proof}{104}
\contentsline {paragraph}{Step 1}{104}
\contentsline {paragraph}{Step 2}{104}
\contentsline {section}{\numberline {5.10}Pappus' theorem}{105}
\contentsline {paragraph}{Proof}{106}
\contentsline {section}{\numberline {5.11}Principle of duality}{107}
\contentsline {section}{\numberline {5.12}Axiomatic projective geometry}{108}
\contentsline {paragraph}{Introducing coordinates in axiomatic projective planes}{109}
\contentsline {paragraph}{Flavour of proof}{109}
\contentsline {section}{Exercises to Chapter\nobreakspace {}5}{110}
\contentsline {chapter}{\numberline {6}Geometry and group theory}{115}
\contentsline {section}{\numberline {6.1}Transformations form a group}{117}
\contentsline {paragraph}{Proof}{117}
\contentsline {section}{\numberline {6.2}Transformation groups}{117}
\contentsline {paragraph}{Discussion}{118}
\contentsline {paragraph}{Example 0. ``No structure''}{118}
\contentsline {paragraph}{Example 1. Euclidean motions}{118}
\contentsline {paragraph}{Example 1a. Symmetry groups}{118}
\contentsline {paragraph}{Example 2. Linear maps}{118}
\contentsline {section}{\numberline {6.3}Klein's Erlangen program}{119}
\contentsline {section}{\numberline {6.4}Conjugacy in transformation groups}{120}
\contentsline {paragraph}{Question:}{120}
\contentsline {paragraph}{Answer:}{120}
\contentsline {paragraph}{Example\nobreakspace {}1. Transpositions in $S_n$}{120}
\contentsline {paragraph}{Proof}{120}
\contentsline {paragraph}{Example\nobreakspace {}2. Fixed point}{121}
\contentsline {paragraph}{Example\nobreakspace {}3. Rotation}{121}
\contentsline {paragraph}{Example\nobreakspace {}4. Translation}{121}
\contentsline {paragraph}{Remark}{121}
\contentsline {section}{\numberline {6.5}Applications of conjugacy}{122}
\contentsline {subsection}{\numberline {6.5.1}Normal forms}{122}
\contentsline {paragraph}{Remark}{123}
\contentsline {subsection}{\numberline {6.5.2}Finding generators}{124}
\contentsline {paragraph}{Example\nobreakspace {}1. How to walk a wardrobe}{124}
\contentsline {paragraph}{Example\nobreakspace {}2. The $15$-puzzle}{124}
\contentsline {paragraph}{Proof}{125}
\contentsline {paragraph}{Step 2}{125}
\contentsline {paragraph}{Step 3}{125}
\contentsline {paragraph}{End of proof}{125}
\contentsline {subsection}{\numberline {6.5.3}The algebraic structure of transformation groups}{125}
\contentsline {paragraph}{Proof}{126}
\contentsline {paragraph}{Remarks}{127}
\contentsline {section}{\numberline {6.6}Discrete reflection groups}{128}
\contentsline {paragraph}{Example 1. Kaleidoscope}{128}
\contentsline {paragraph}{Example 2. Barber's shop}{128}
\contentsline {paragraph}{Example 3. Mus\'ee Gr\'evin}{128}
\contentsline {section}{Exercises to Chapter\nobreakspace {}6}{129}
\contentsline {chapter}{\numberline {7}Topology}{133}
\contentsline {paragraph}{``Point-set topology''}{133}
\contentsline {paragraph}{``Rubber-sheet geometry''}{133}
\contentsline {section}{\numberline {7.1}Definition of a topological space}{134}
\contentsline {paragraph}{Example\nobreakspace {}1}{134}
\contentsline {paragraph}{Example\nobreakspace {}2}{134}
\contentsline {paragraph}{Example\nobreakspace {}3}{135}
\contentsline {section}{\numberline {7.2}Motivation from metric spaces}{135}
\contentsline {paragraph}{Definition}{135}
\contentsline {paragraph}{Equivalent conditions}{135}
\contentsline {section}{\numberline {7.3}Continuous maps and homeomorphisms}{137}
\contentsline {subsection}{\numberline {7.3.1}Definition of a continuous map}{137}
\contentsline {paragraph}{Example\nobreakspace {}1}{137}
\contentsline {paragraph}{Example\nobreakspace {}2}{137}
\contentsline {subsection}{\numberline {7.3.2}Definition of a homeomorphism}{138}
\contentsline {paragraph}{Example\nobreakspace {}3}{138}
\contentsline {paragraph}{Example\nobreakspace {}4}{139}
\contentsline {paragraph}{Example\nobreakspace {}5}{139}
\contentsline {subsection}{\numberline {7.3.3}Homeomorphisms and the Erlangen program}{139}
\contentsline {subsection}{\numberline {7.3.4}The homeomorphism problem}{140}
\contentsline {section}{\numberline {7.4}Topological properties}{140}
\contentsline {subsection}{\numberline {7.4.1}Connectedness}{141}
\contentsline {paragraph}{Proof}{141}
\contentsline {paragraph}{Remark}{142}
\contentsline {subsection}{\numberline {7.4.2}Compactness}{142}
\contentsline {paragraph}{Example}{143}
\contentsline {subsection}{\numberline {7.4.3}Continuous image of a compact space is compact}{144}
\contentsline {paragraph}{Proof}{144}
\contentsline {subsection}{\numberline {7.4.4}An application of topological properties}{144}
\contentsline {section}{\numberline {7.5}Subspace and quotient topology}{145}
\contentsline {paragraph}{Proof}{146}
\contentsline {section}{\numberline {7.6}Standard examples of glueing}{146}
\contentsline {paragraph}{Example\nobreakspace {}0}{146}
\contentsline {paragraph}{Example\nobreakspace {}1}{146}
\contentsline {paragraph}{Example\nobreakspace {}2}{147}
\contentsline {paragraph}{Example\nobreakspace {}3}{147}
\contentsline {paragraph}{Example\nobreakspace {}4}{147}
\contentsline {paragraph}{Step\nobreakspace {}1: Main claim}{149}
\contentsline {paragraph}{Step\nobreakspace {}2}{149}
\contentsline {paragraph}{Step\nobreakspace {}3}{149}
\contentsline {section}{\numberline {7.7}Topology of $\PP ^n_\@mathbb R$}{150}
\contentsline {section}{\numberline {7.8}Nonmetric quotient topologies}{151}
\contentsline {paragraph}{Example\nobreakspace {}1 (the mousetrap topology)}{151}
\contentsline {paragraph}{Example\nobreakspace {}2: Quadratic forms}{152}
\contentsline {section}{\numberline {7.9}Basis for a topology}{153}
\contentsline {paragraph}{Proof}{154}
\contentsline {paragraph}{Example 1}{154}
\contentsline {paragraph}{Example 2}{154}
\contentsline {paragraph}{Example 3: Profinite topology of an infinite group}{154}
\contentsline {paragraph}{Remark}{155}
\contentsline {section}{\numberline {7.10}Product topology}{155}
\contentsline {section}{\numberline {7.11}The Hausdorff property}{156}
\contentsline {paragraph}{Example 1}{156}
\contentsline {paragraph}{Example 2}{156}
\contentsline {paragraph}{Example 3}{156}
\contentsline {paragraph}{Proof}{157}
\contentsline {section}{\numberline {7.12}Compact versus closed}{157}
\contentsline {paragraph}{Proof}{157}
\contentsline {section}{\numberline {7.13}Closed maps}{158}
\contentsline {paragraph}{Example 1}{158}
\contentsline {paragraph}{Proof}{158}
\contentsline {paragraph}{Example 2}{158}
\contentsline {paragraph}{Proof}{160}
\contentsline {section}{\numberline {7.14}A criterion for homeomorphism}{160}
\contentsline {paragraph}{Proof}{160}
\contentsline {paragraph}{Example}{160}
\contentsline {section}{\numberline {7.15}Loops and the winding number}{160}
\contentsline {paragraph}{Question}{160}
\contentsline {paragraph}{Answer}{161}
\contentsline {subsection}{\numberline {7.15.1}Paths, loops and families}{161}
\contentsline {paragraph}{Tentative definition}{161}
\contentsline {paragraph}{Remark}{162}
\contentsline {paragraph}{Proof}{162}
\contentsline {paragraph}{Definition}{163}
\contentsline {paragraph}{Example}{163}
\contentsline {subsection}{\numberline {7.15.2}The winding number}{163}
\contentsline {paragraph}{Definition}{164}
\contentsline {paragraph}{Proof}{164}
\contentsline {subsection}{\numberline {7.15.3}Winding number is constant in a family}{166}
\contentsline {paragraph}{Proof}{166}
\contentsline {subsection}{\numberline {7.15.4}Applications of the winding number}{167}
\contentsline {paragraph}{Proof}{167}
\contentsline {paragraph}{Remark}{167}
\contentsline {paragraph}{Proof}{167}
\contentsline {section}{Exercises to Chapter\nobreakspace {}7}{168}
\contentsline {chapter}{\numberline {8}Geometry of transformation groups}{175}
\contentsline {section}{\numberline {8.1}Topology on groups}{176}
\contentsline {paragraph}{Example 1}{177}
\contentsline {paragraph}{Example 2}{177}
\contentsline {paragraph}{Example 3}{177}
\contentsline {paragraph}{Example 4}{177}
\contentsline {paragraph}{Example 5}{178}
\contentsline {section}{\numberline {8.2}Dimension counting}{178}
\contentsline {paragraph}{Proof}{179}
\contentsline {section}{\numberline {8.3}Compact and noncompact groups}{180}
\contentsline {paragraph}{Proof}{180}
\contentsline {paragraph}{Example 1}{181}
\contentsline {paragraph}{Example 2}{181}
\contentsline {paragraph}{Example 3}{181}
\contentsline {paragraph}{Discussion}{182}
\contentsline {paragraph}{Proof}{182}
\contentsline {section}{\numberline {8.4}Components}{182}
\contentsline {paragraph}{Remark}{182}
\contentsline {paragraph}{Proof}{182}
\contentsline {section}{\numberline {8.5}The geometry of $\SO (n)$}{183}
\contentsline {subsection}{\numberline {8.5.1}Quaternions}{184}
\contentsline {paragraph}{Proof}{185}
\contentsline {paragraph}{Remark}{185}
\contentsline {subsection}{\numberline {8.5.2}Quaternions and rotations}{186}
\contentsline {paragraph}{Proof}{186}
\contentsline {subsection}{\numberline {8.5.3}Spheres and special orthogonal groups}{187}
\contentsline {paragraph}{Proof}{187}
\contentsline {paragraph}{Remark}{188}
\contentsline {section}{\numberline {8.6}The group $\SU (2)$}{188}
\contentsline {paragraph}{Remark}{188}
\contentsline {paragraph}{Proof}{189}
\contentsline {section}{\numberline {8.7}The electron spin in quantum mechanics}{189}
\contentsline {subsection}{\numberline {8.7.1}The story of the electron spin}{189}
\contentsline {subsection}{\numberline {8.7.2}Measuring spin: the Stern--Gerlach device}{191}
\contentsline {subsection}{\numberline {8.7.3}The spin operator}{193}
\contentsline {subsection}{\numberline {8.7.4}Rotate the device}{194}
\contentsline {subsection}{\numberline {8.7.5}The solution}{195}
\contentsline {section}{\numberline {8.8}Preview of Lie groups}{196}
\contentsline {section}{Exercises to Chapter\nobreakspace {}8}{197}
\contentsline {chapter}{\numberline {9}Concluding remarks}{203}
\contentsline {section}{\numberline {9.1}On the history of geometry}{203}
\contentsline {subsection}{\numberline {9.1.1}Greek geometry and rigour}{203}
\contentsline {subsection}{\numberline {9.1.2}The parallel postulate}{204}
\contentsline {subsection}{\numberline {9.1.3}Coordinates versus axioms}{207}
\contentsline {section}{\numberline {9.2}Group theory}{208}
\contentsline {subsection}{\numberline {9.2.1}Abstract groups versus transformation groups}{208}
\contentsline {subsection}{\numberline {9.2.2}Homogeneous and principal homogeneous spaces}{209}
\contentsline {paragraph}{Definition}{209}
\contentsline {paragraph}{Definition}{210}
\contentsline {paragraph}{Example}{210}
\contentsline {subsection}{\numberline {9.2.3}The Erlangen program revisited}{210}
\contentsline {subsection}{\numberline {9.2.4}Affine space as a torsor}{211}
\contentsline {section}{\numberline {9.3}The famous trichotomy}{212}
\contentsline {section}{\numberline {9.4}Geometry in physics}{214}
\contentsline {subsection}{\numberline {9.4.1}The Galilean group and Newtonian dynamics}{215}
\contentsline {subsection}{\numberline {9.4.2}The Poincar\'e group and special relativity}{216}
\contentsline {subsection}{\numberline {9.4.3}Wigner's classification: elementary particles}{218}
\contentsline {subsection}{\numberline {9.4.4}The Standard Model and beyond}{219}
\contentsline {subsection}{\numberline {9.4.5}Other connections}{220}
\contentsline {section}{\numberline {9.5}Further directions}{220}
\contentsline {chapter}{\numberline {A}Metrics}{223}
\contentsline {paragraph}{Definition}{223}
\contentsline {paragraph}{Definition}{224}
\contentsline {paragraph}{Definition}{224}
\contentsline {section}{Exercises to Appendix\nobreakspace {}A}{224}
\contentsline {chapter}{\numberline {B}Linear algebra}{227}
\contentsline {section}{\numberline {B.1}Bilinear form and quadratic form}{227}
\contentsline {paragraph}{Definition}{227}
\contentsline {section}{\numberline {B.2}Euclid and Lorentz}{228}
\contentsline {section}{\numberline {B.3}Complements and bases}{229}
\contentsline {paragraph}{Definition}{229}
\contentsline {paragraph}{Proof}{229}
\contentsline {paragraph}{Proof}{230}
\contentsline {section}{\numberline {B.4}Symmetries}{231}
\contentsline {paragraph}{Proof}{231}
\contentsline {paragraph}{Proof}{231}
\contentsline {paragraph}{Proof}{232}
\contentsline {section}{\numberline {B.5}Orthogonal and Lorentz matrixes}{232}
\contentsline {section}{\numberline {B.6}Hermitian forms and unitary matrixes}{232}
\contentsline {section}{Exercises to Appendix\nobreakspace {}B}{234}
\contentsline {chapter}{References}{235}
\contentsline {chapter}{Index}{237}