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Alexandrov Geometry, Angles Defined As in Section 5.3 Always Exist (See Theorem 7.3.1C and Corollary 8.3.2B)

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Alexandrov geometry preliminary version no. 1 S. Alexander, V. Kapovitch, A. Petrunin arXiv:1903.08539v1 [math.DG] 20 Mar 2019 ii Contents I Preliminaries 1 1 Model plane 3 1.1 Trigonometry ........................ 3 1.2 Hemispherelemma ..................... 6 2 Metric spaces 9 2.1 Metrics and their relatives . 9 2.2 Notations .......................... 11 2.3 Lengthspaces........................ 13 2.4 Convexsets ......................... 16 2.5 Quotientspaces....................... 17 2.6 Gluing and doubling . 19 2.7 Kuratowskyembedding . 20 3 Ultralimits 21 3.1 Ultrafilters ......................... 21 3.2 Ultralimits of points . 22 3.3 Ultralimits of spaces . 24 3.4 Ultratangentspace . 26 3.5 Gromov–Hausdorff convergence and ultralimits . 27 3.6 Ultralimits of sets . 29 3.7 Ultralimits of functions . 29 4 Maps and functions 31 4.1 Submaps........................... 31 4.2 Lipschitz conditions . 32 4.3 Isometriesandsumbetries . 33 4.4 Speedofcurves ....................... 35 4.5 Convexreal-to-realfunctions . 36 4.6 Convexfunctionsonametricspace. 38 1 2 CONTENTS 5 The ghost of Euclid 41 5.1 Geodesics, triangles and hinges . 41 5.2 Modelanglesandtriangles. 42 5.3 Anglesandthefirstvariation . 44 5.4 Constructions........................ 46 5.5 Spaceofdirectionsandtangentspace . 47 5.6 Velocityofcurves...................... 47 5.7 Differential ......................... 49 5.8 Remarks........................... 50 6 Dimension theory 51 6.1 Definitions.......................... 51 6.2 Lineardimension ...................... 53 II Definitions and constructions 55 7 Definitionsofcurvatureboundedbelow 57 7.1 Four-pointcomparison . 57 7.2 Geodesics .......................... 60 7.3 Morecomparisons ..................... 62 7.4 Functioncomparison . 65 7.5 Development ........................ 66 7.6 Local definitions and globalization . 70 7.7 Propertiesofgeodesicsandangles . 76 7.8 Onpositivelowerbound . 81 7.9 Remarksandopenproblems. 84 8 Definitionsofcurvatureboundedabove 85 8.1 Four-pointcomparison.. 85 8.2 Geodesics .......................... 86 8.3 Morecomparisons ..................... 88 8.4 Thintriangles........................ 91 8.5 Functioncomparison . 94 8.6 Convexsets ......................... 95 8.7 Development ........................ 98 8.8 Patchwork globalization . 99 8.9 Reshetnyak’sgluingtheorem . .101 8.10 Spaceofgeodesics . .103 8.11 Lifting globalization . 107 8.12 Reshetnyakmajorization. .110 8.13 Hadamard–Cartantheorem . .118 8.14Angles............................121 8.15 Remarksandopenproblems. .123 CONTENTS 3 9 Kirszbraun revisited 125 9.1 Shortmapextensiondefinitions. .125 9.2 (1+n)-pointcomparison . .129 9.3 Helly’stheorem.......................131 9.4 Kirszbraun’stheorem. .132 9.5 Curvaturefree .......................138 9.6 Remarksandopenproblems. .139 10 Warped products 141 10.1Definition ..........................141 10.2 Extendeddefinition. .144 10.3Examples ..........................144 10.4 1-dimensionalbase . .145 10.5 Generalcase.. .. .. .. .. .. .. .. .. .. ..147 III Structureandtools 149 11Firstorderdifferentiation 151 11.1 Intrinsic propertyoftangentspace . .151 11.2Differential .........................154 11.3 Ultratangentspace . .156 11.4Gradient...........................156 11.5Polarvectors ........................162 11.6 Linearsubspaceoftangentspace . .163 11.7Comments..........................165 12DimensionofCATspaces 167 12.1 The caseofcomplete geodesicspaces . .167 12.2 ThecaseofCATspaces . .172 12.3 DimensionofCBAspaces . .176 12.4 Finite-dimensional CAT spaces . 179 12.5 Remarksandopenproblems. .181 13DimensionofCBBspaces 183 13.1 Strutsandrank. .183 13.2 Right-inversetheorem . .184 13.3 DimensiontheoremforCBB . .187 13.4 Finite dimensional CBB spaces . 191 13.5 One-dimensionalCBBspaces . .194 4 CONTENTS 14 Gradient flow 197 14.1 Gradient-likecurves . .197 14.2 Gradientcurves. .200 14.3Gradientflow ........................207 14.4 Line splitting theorem . 208 14.5Radialcurves . .. .. .. .. .. .. .. .. .. ..210 14.6 Radialcomparisons. .212 14.7 Gradientexponent . .217 14.8Remarks...........................218 Bibliography 229 Part I Preliminaries 1 Chapter 1 Model plane 1.1 Trigonometry Given a real number κ, the model κ-plane will be a complete sim- ply connected 2-dimensional Riemannian manifold of constant curva- ture κ. The model κ-plane will be denoted by M2(κ). If κ > 0, M2(κ) is isometric to a sphere of radius 1 ; the ⋄ √κ unit sphere M2(1) will be also denoted by S2. If κ = 0, M2(κ) is the Euclidean plane, which is also de- ⋄ noted by E2 If κ < 0, M2(κ) is the Lobachevsky plane with curvature ⋄ κ. Set ̟κ = diam M2(κ), so ̟κ = if κ 6 0 and ̟κ = π/√κ if κ > 0. ∞ The distance between points x, y M2(κ) will be denoted by x y , and [xy] will denote the segment connecting∈ x and y. The segment| − | [xy] is uniquely defined for κ 6 0 and for κ > 0 it is defined uniquely if x y <̟κ = π/√κ. | − | A triangle in M2(κ), with vertices x,y,z will be denoted by [xyz]. Formally, a triangle is an ordered set of its sides, so [xyz] is just a short notation for a triple ([yz], [zx], [xy]). y The angle of [xyz] at x will be denoted by ∡[x z ]. By ˜ κ a,b,c we denote a triangle in M2(κ) with sidelengths a,b,c, so△ [{xyz] =} ˜ κ a,b,c means that x,y,z M2(κ) are such that △ { } ∈ x y = c, y z = a, z x = b. | − | | − | | − | For ˜ κ a,b,c to be defined, the sides a,b,c must satisfy the triangle △ { } 3 4 CHAPTER 1. MODEL PLANE inequality. If κ > 0, we require in addition that a + b + c < 2 ̟κ; otherwise ˜ κ a,b,c is considered to be undefined. · △ { } Trigonometric functions. We will need three “trigonometric func- tions” in M2(κ): csκ, snκ and mdκ; cs stands for cosine, sn stands for sine and md, for modified distance. They are defined as the solutions of the following initial value prob- lems respectively: x′′ + κ x =0, y′′ + κ y =0, z′′ + κ z =1, · · · x(0) = 1, y(0) = 0, z(0) = 0, x′(0) = 0. y′(0) = 1. z′(0) = 0. Namely we set csκ(t)=x(t), snκ(t)= y(t) and κ κ z(t) if 0 6 t 6 ̟ , md (t)= 2 κ ( κ if t>̟ . Here are the tables which relate our trigonometric functions to the standard ones, where we take κ > 0: κ 1 1 κ 1 sn± = sn± (x √κ); cs± = cs± (x √κ); √κ · · · 1 1 sn− x = sinh x; cs− x = cosh x; sn0x = x; cs0x = 1; sn1x = sin x; cs1x = cos x. κ 1 1 md± = md± (x √κ); κ · · 1 md− x = cosh x 1; − md0x = 1 x2; 2 · 1 cos x for x 6 π, md1x = − " 2 for x> π. Note that x b a mdκ(x)= snκ(x) dx for x 6 ̟κ = g w · ˜ 0 κ κ ϕ =∡ ˜ a; b, c { { } ϕ κ ; Let ϕ be the angle of ˜ a,b,c opposite b, c △ { } to a. In this case we will write c } κ a = g˜ ϕ; b,c or ϕ =˜∡κ a; b,c . { } { } 1.1. TRIGONOMETRY 5 κ The functions g˜ and∡ ˜ κ will be called cor- respondingly the model side and the model angle. Set κ κ κ g˜ ϕ; b, c = g˜ ϕ; b,c := g˜ π ϕ; b,c , { − } { − } { − } κ this way we define g˜ ϕ; b,c for some negative values of b and c. { } 1.1.1. Properties of standard functions. a) For fixed a and ϕ, the function κ y(t) = mdκ g˜ ϕ; a,t { } satisfies the following differential equation: y′′ + κ y =1. · b) Let α:[a,b] M2(κ) be a unit-speed geodesic, and A be the image of→ a complete geodesic. If f(t) is the distance from α(t) to A, the function y(t)=snκ(f(t)) satisfies the following differential equation: y′′ + κ y =0 · for y =0. c) For fixed6 κ, b and c, the function a ∡˜ κ a; b,c 7→ { } is increasing and defined on a real interval. Equivalently, the function κ ϕ g˜ ϕ; b,c 7→ { } is increasing and defined if b,c<̟κ and ϕ [0, π]1. d) For fixed ϕ,a,b,c, the functions ∈ κ κ ∡˜ κ a; b,c and κ g˜ ϕ; b,c 7→ { } 7→ { } are respectively non-decreasing (in fact, decreasing, if a / b c ,b + c ) and non-increasing (in fact, increasing,∈ if ϕ{| /− 0|, π ). } e) (Alexandrov’s∈{ } lemma) Assume that for real numbers a, b, a′, b′, x and κ, the following two expressions are defined 1Formally speaking, if κ > 0 and b + c > ̟κ, it is defined only for ϕ ∈ [0, π), but g˜ κ{ϕ; b, c} can be extended to [0, π] as a continuos function. 6 CHAPTER 1. MODEL PLANE κ κ 1. ∡˜ a; b, x +˜∡ a′; b′, x π, κ{ } { κ }− 2. ∡˜ a′; b + b′,a ∡˜ x; a,b , { }− { } Then they have the same sign. All the properties except Alexandrov’s lemma (e) can be shown by direct calculation. Alexandrov’s lemma is reformulated in 5.2.1 and is proved there. Cosine law. The above formulas easily imply the cosine law in M2(κ), which can be expressed as follows b2 + c2 a2 − if κ =0 cos ϕ = 2 b c csκa · ·csκb csκc csκa csκb csκc − · if κ =0 − · if κ =0 κ snκb snκc 6 κ snκb snκc 6 · · · · However, rather than using these explicit formulas, we mainly will κ use the properties of∡ ˜ κ and g˜ listed in 1.1.1. 1.2 Hemisphere lemma 1.2.1. Hemisphere lemma. For κ > 0, any closed path of length < 2 ̟κ (respectively, 6 2 ̟κ) in M2(κ) lies in an open (respectively, closed)· hemisphere. · Proof. By rescaling, we may assume that κ = 1 and thus ̟κ = π and M2(κ)= S2. Let α be a closed curve in S2 of length 2 ℓ. Assume ℓ< π. Let αˇ be a subarc of α of length ℓ, with· endpoints p and q. Since p q 6 ℓ< π, there is a unique geodesic [pq] in S2. Let z be the midpoint| − of| [pq]. We claim that α lies in the open hemisphere centered at z. If not, α intersects the boundary great circle in a point, say r. Without loss of generality we may assume that r αˇ.
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  • Metric Foundations of Geometry. I

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    METRIC FOUNDATIONS OF GEOMETRY. I BY GARRETT BIRKHOFF 1. Introduction. It is shown below by elementary methods that «-dimen- sional Euclidean, spherical, and hyperbolic geometry can be characterized by the following postulates. Postulate I. Space is metric (in the sense of Fréchet). Postulate II. Through any two points a line segment can be drawn, locally uniquely. Postulate III. Any isometry between subsets of space can be extended to a self-isometry of all space. The outline of the proof has been presented elsewhere (G. Birkhoff [2 ]('))• Earlier, H. Busemann [l] had shown that if, further, exactly one straight line connected any two points, and if bounded sets were compact, then the space was Euclidean or hyperbolic; moreover for this, Postulate III need only be assumed for triples of points. Later, Busemann extended the result to elliptic space, assuming that exactly one geodesic connected any two points, that bounded sets were compact, and that Postulate III held locally for triples of points (H. Busemann [2, p. 186]). We recall Lie's celebrated proof that any Riemannian variety having local free mobility is locally Euclidean, spherical, or hyperbolic. Since our Postu- late II, unlike Busemann's straight line postulates, holds in any Riemannian variety in which bounded sets are compact, our result may be regarded as freeing Lie's theorem for the first time from its analytical hypotheses and its local character. What is more important, we use direct geometric arguments, where Lie used sophisticated analytical and algebraic methods(2). Our methods are also far more direct and elementary than those of Busemann, who relies on a deep theorem of Hurewicz.
  • Kirszbraun's Theorem

    Kirszbraun's Theorem

    Version of 14.9.18 Kirszbraun’s theorem D.H.Fremlin University of Essex, Colchester, England Wikipedia gives a statement of this theorem and an outline of its history, but no online source for the proof. It’s so pretty that I write one out here. 1 The essential ideas 1A The context I’ll come to the actual statement of the theorem, taken from http://en.wikipedia.org/ wiki/Kirszbraun theorem, in 1G below. This will be in the standard full-generality form concerning Lipschitz maps between (real) Hilbert spaces H1 and H2. If you aren’t familiar with ‘Hilbert spaces’ (http://en.wikipedia.org/wiki/Hilbert space), then you will probably prefer to start by taking both n H1 and H2 to be a Euclidean space R , with the inner product n (x y)= x.y = ξiηi | Pi=1 n if x =(ξ1,... ,ξn) and y =(η1,... ,ηn) belong to R , and the norm x = (x x)= n ξ2, k k p | pPi=1 i so that n 2 x y = (ξi ηi) k − k pPi=1 − is the Euclidean distance from x to y calculated with the n-dimensional version of Pythagoras’ theorem. In fact all the really interesting ideas of the proof, in 1B-1F below, are needed for the two-dimensional case n = 2. (The case n = 1 is much easier, as noted in Wikipedia.) I do have to warn you, however, that ordinary proofs of Kirszbraun’s theorem involve ‘Tychonoff’s theorem’ (see 2B below), and unless the words ‘topology’, ‘compact set’ and ‘continuous function’ mean something to you, you are going to have a good deal of work to do to understand the ‘first proof’ offered in 2.