Appendix A

Notations and symbols

Theorems, propositions, and lemmata are numbered consecutively in each chap- ter, so that Lemma 1 may be followed by Proposition 2 and that by Theorem 3. Chapters are subdivided into sections but numbering of formulas is within chap- ters, not sections. The end of a proof is indicated by
.Thesymbols:=or=: mean that the side of the equation, where the colon is, is defined by the other side. Sometimes provided is used as an abbreviation for if and only if.

A ⇒ B means A implies B, A ⇔ B is defined by (A ⇒ B)and(B ⇒ A), ∀ abbreviates forall.

Moreover,

∀x∈S A (x) ⇒ B (x)meansA (x) implies B (x) for all x ∈ S, ∃, there exist(s) and f : A → B that f is a mapping from A into B. If f is a mapping from B into C and g a mapping from A into B,thenfg : A → C is defined by (fg)(x):=f [g (x)] for all x ∈ A. If f is a mapping from A into B and if H is a subset of M,thenf | H (the so– called restriction of f on H) denotes the mapping ϕ : H → B with ϕ (x):=f (x) for all x ∈ H. If S isaset,thenid:S → S designates the mapping defined by id (x)=x for all x ∈ S. If S is a set, then {x ∈ S | P (x)} denotes the set of all x in S which satisfy property P . 232 Appendix A. Notations and symbols

If A, B are sets, then A\B := {x ∈ A | x ∈ B}. R denotes the set of all real numbers, furthermore,

R≥0 := {x ∈ R | x ≥ 0},

R>0 := R≥0\{0}.

If A1,...,An are sets, their cartesian product is

A1 × A2 ×···×An := {(x1,...,xn) | xi ∈ Ai for i =1,...,n}.

If M is a set, #M designates its cardinality. √ If a is a non–negative real number, a denotes the real number b ≥ 0 satisfying b2 = a. Appendix B

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Index

Action of a group 16 direction of a angle 60 translation group 11 angle measure 60f distance between a point angles of parallelism 61f and a hyperplane 54 axiom of coincidence 38 distance function axis of a – elliptic 208, 216 translation group 11 – euclidean 20 – hyperbolic 20, 126f – Lorentz–Minkowskian 172, 175, 194, 196 Ball 45f, 217 –ofX0 := X\{0} 215 bundle 148 – spherical 208, 216 Cauchy’s functional equation 24 Einstein’s causal automorphism 177, 180 – cylinder universe 197 Cayley–Klein model 66f – cylindrical world 197 characterization of Einstein distance 197 translations 78 elliptic circular helix 200 – geometry 214 closed line 205 – group 214 contact 135 – motion 214 contact relation of – points 211 Lie sphere geometry 150f end of a line 56f cosine theorem of end of a ray 59 hyperbolic geometry 60 equidistant surface 54 cross ratio 66, 118ff ES–space 210 cycle coordinates euclidean – of Laguerre cycles 135 –distance 20 – of Lie cycles 154 –geometry 20 cyclographic projection 141 – hyperplane 48f cylinder model 142f – line 39 – metric space 38 – motion 31, 33 event 175 e Sitter distance 205 D example of a quasi– de Sitter’s world 205 hyperplane which is diffeomorphism not a hyperplane 50f (line preserving) 174 examples of real inner dilatation 153 product spaces 2f, 50 dimension–free ix, xif 240 Appendix C. Index

Fourty five degree isometries hyperplane 142 –ofametricspace 75 functional equation – of an ES–space 212 of hyperbolic sine 30 –ofC (Z) 198 functional equation of –ofE (X) 212 Blumenthal lines 38ff –ofS (X) 212 functional equation –of(X, eucl) 75f, 80 of Cauchy 24 –of(X, hyp) 75f, 80 functional equation isomorphic geometries 16f of Jensen 6 isomorphic real inner functional equation product spaces 1f of translations 11 functional equation of ensen’s functional equation 6 2–point invariants 203, 207 J fundamental theorem of Lie sphere geometry 173 Kernel of a fundamental theorem of translation group 12 M¨obius sphere geometry 98 Klein’s Erlangen Programme ixf future of an event 187 Laguerre cycle 134 Generator 142 Laguerre sphere geometry 151 geometrical subspace 63 Laguerre transformation 151f, 163 geometry of a group Lag (X) 151, 170 of permutations 16 Lag∗(X) 151 geometry of de Sitter’s Lie cycle 150 world 205 Lie quadric 155 geometry of Einstein’s Lie sphere geometry 151 cylindrical world 200 Lie transformation 151 Lie (X) 151 Homogeneous coordinates 136, 154 lines in the sense horocycle 62f – of Blumenthal 38f horocycle as paraboloid 62f –ofMenger 43 hyperbolic lines – coordinates 74 – of de Sitter’s world 205 – distance 20, 126ff – of Einstein’s universe 200 – geometry 20, 125f light cone 186f –hyperplane 49 lightlike hyperplane 188ff, 191, 194 – line 39 lightlike lines 185, 187 – metric space 38 Lorentz boost 160f, 163f, 167, 172, – midpoint 81 – 176, 179ff, 182ff, 227f – motion 31, 33 Lorentz group 172 – subspaces 49f Lorentz–Minkowski geometry 172 hypercycle 54 Lorentz transformation 172, 175 hyperellipsoid 47 Lorentz transformations hyperplane 48f as Lie transformations 193 hyperplane cut 142 Maximal subspace 50 Improper Lorentz boost 160 measure of an angle 60f inequality of Cauchy–Schwarz 4 Menger interval 43 integral equation 96ff metric space 37f invariant 16 mid–cycle 141 invariant notion 16 midpoint 81 inversion 94 mild–hypotheses involution 95, 102 characterizations 7, 80, 170, 172ff 241

Mn–sphere 112 Ray 59 M n–sphere 115 ray through an end 59 M¨obius ball 93 real inner product space 1 M¨obius circle 111 reflection 94 M¨obius group 93 relativistic addition 181 M¨obius sphere geometry 93 M¨obius transformation 93 Separable M¨obius transformations translation group 14f as Lie transformations 167ff separated points 205 motion separation 139 – elliptic 214 sides – euclidean 31, 33 – of a ball 123 – hyperbolic 31, 33 – of a hyperplane 133 –ofametricspace 76 –ofanM–ball 123 – of de Sitter’s world 205 similitude 93 – of Einstein’s cylinder universe 200 spacelike line 185f – of Lorentz–Minkowski geometry 176 spear 133 – spherical 214 spear coordinates 135 spherical Norm 5 – geometry 214 n–plane 112 – group 214 null–lines 200, 205 – motion 214 – points 211 – subspace 50 Open line 205 spherically independent 114 orthochronous Lorentz stabilizer 35 transformations 179 stereographic projection 121 orthogonal group 7 strongly independent 165f orthogonal mapping 5 subspace 49, 217 orthogonality 51, 107, 116 suitable hyperplane 121f symmetry axiom 38 Parallelity 59, 138, 142 parametric Tangential representation 54ff, 97 – distance 140 past of an event 187 – hyperplane 110, 134, 190 pencils theorem of Pythagoras – elliptic 146 – euclidean case 51 – hyperbolic 147 – hyperbolic case 51 – parabolic 144 time axis 175 periodic lines 218ff timelike lines 185ff permutation product 10 transformation formulas 70, 72f Perm X 10 translation equation 11 Poincar´e’s model 126, 131 translation group 11 point of contact 135 triangle inequality 5, 38 power 139 two–point invariants 202f, 205, 207 projective – geometry 157 – group 157 Uniform characterization of – transformation 157 euclidean and hyperbolic proper Lorentz boost 160 geometry 21 ... 33 proper Laguerre unit ball 95 sphere geometry 151 universe (cylinder universe) 197

Quasi–hyperplane 50f Velocity of signals 185 242 Appendix C. Index

Weierstrass coordinates 229 Weierstrass map 66 Weierstrass model 131 world line 185

X, occasionally also (X, δ), as standard notation for a real inner product space containing two linearly independent elements 1, 5

Y as standard notation for Y = X ⊕ R with product (3.54) 120, 142ff

Z as standard notation for Z = X ⊕ R with product (3.88) 141, 144, 175