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.