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Pure Metric Geometry Pure metric geometry: introductory lectures Anton Petrunin arXiv:2007.09846v4 [math.MG] 20 Sep 2021 We discuss only domestic affairs of metric spaces; applications are given only as illustrations. These notes could be used as an introductory part to virtually any course in metric geometry. It is based on a part of the course in metric geometry given at Penn State, Spring 2020. The complete lectures can be found on the author’s website; it includes an introduction to Alexandrov geometry based on [1] and metric geometry on manifolds [28] based on a simplified proof of Gromov’s systolic inequality given by Alexander Nabutovsky [22]. A part of the text is a compilation from [1, 2, 24, 26, 27] and its drafts. I want to thank Sergio Zamora Barrera for help. Contents 1 Definitions 5 A Metricspaces ........................ 5 B Variationsofdefinition. 6 C Completeness ........................ 7 D Compactspaces....................... 8 E Properspaces........................ 9 F Geodesics .......................... 9 G Geodesicspacesandmetrictrees . 9 H Length............................ 10 I Lengthspaces........................ 11 2 Universal spaces 15 A Embeddinginanormedspace. 15 B Extensionproperty.. .. .. .. .. .. .. .. .. 17 C Universality......................... 19 D Uniquenessandhomogeneity . 20 E Remarks........................... 22 3 Injective spaces 23 A Admissible and extremal functions . 23 B Injectivespaces ....................... 25 C Spaceofextremalfunctions . 27 D Injectiveenvelope. .. .. .. .. .. .. .. .. .. 29 E Remarks........................... 30 4 Space of sets 31 A Hausdorffdistance ..................... 31 B Hausdorffconvergence . 32 C Anapplication ....................... 34 D Remarks........................... 35 3 5 Space of spaces 37 A Gromov–Hausdorffmetric . 37 B Approximations....................... 39 C Almostisometries. .. .. .. .. .. .. .. .. .. 39 D Convergence......................... 41 E Uniformly totally bonded families . 42 F Gromov’sselectiontheorem . 43 G Universalambientspace . 45 H Remarks........................... 46 6 Ultralimits 49 A Facesofultrafilters .. .. .. .. .. .. .. .. .. 49 B Ultralimitsofpoints . 50 C Ultralimitsofspaces . 52 D Ultrapower ......................... 53 E Tangentandasymptoticspaces . 54 F Remarks........................... 55 A Semisolutions 57 Bibliography 79 Lecture 1 Definitions In this lecture we give some conventions used further and remind some definitions related to metric spaces. We assume some prior knowledge of metric spaces. For a more detailed introduction, we recommend the first couple of chapters in the book by Dmitri Burago, Yuri Burago, and Sergei Ivanov [7]. A Metric spaces The distance between two points x and y in a metric space will be X denoted by x y or x y X . The latter notation is used if we need to emphasize| that− | the| distance− | is taken in the space . X Let us recall the definition of metric. 1.1. Definition. A metric on a set is a real-valued function (x, y) X 7→ x y X that satisfies the following conditions for any three points x,y,z7→ | − | : ∈ X (a) x y > 0, | − |X (b) x y =0 x = y, | − |X ⇐⇒ (c) x y = y x , | − |X | − |X (d) x y + y z > x z . | − |X | − |X | − |X Recall that a metric space is a set with a metric on it. The elements of the set are called points. Most of the time we keep the same notation for the metric space and its underlying set. The function distx : y x y 7→ | − | is called the distance function from x. 5 6 LECTURE 1. DEFINITIONS Given R [0, ] and x , the sets ∈ ∞ ∈ X B(x, R)= y x y < R , { ∈ X | | − | } B[x, R]= y x y 6 R { ∈ X | | − | } are called, respectively, the open and the closed balls of radius R with center x. If we need to emphasize that these balls are taken in the metric space , we write X B(x, R)X and B[x, R]X . 1.2. Exercise. Show that p q + x y 6 p x + p y + q x + q y | − |X | − |X | − |X | − |X | − |X | − |X for any four points p, q, x, and y in a metric space . X B Variations of definition Pseudometris. A metric for which the distance between two distinct points can be zero is called a pseudometric. In other words, to define pseudometric, we need to remove condition (b) from 1.1. The following observation shows that nearly any question about pseudometric spaces can be reduced to a question about genuine metric spaces. Assume is a pseudometric space. Consider an equivalence rela- tion on Xdefined by ∼ X x y x y =0. ∼ ⇐⇒ | − | Note that if x x′, then y x = y x′ for any y . Thus, defines a metric∼ on the quotient| − | set| −/ .| This way we∈ X obtain a metric|∗− ∗| space ′. The space ′ is called theX ∼corresponding metric space for the pseudometricX spaceX . Often we do not distinguish between ′ and . X X X ∞-metrics. One may also consider metrics with values in R ; we might call them -metrics, but most of the time we use the∪ {∞} term metric. ∞ Again nearly any question about -metric spaces can be reduced to a question about genuine metric spaces.∞ Set x y x y < ; ≈ ⇐⇒ | − | ∞ C. COMPLETENESS 7 it defines another equivalence relation on . The equivalence class of a point x will be called the metricX component of x; it will be ∈ X denoted by x. One could think of x as B(x, )X the open ball centered at xX and radius in . X ∞ It follows that any ∞-metricX space is a disjoint union of genuine metric spaces the metric∞ components of the original -metric space. ∞ 1.3. Exercise. Given two sets A and B on the plane, set A B = µ(A B), | − | △ where µ denotes the Lebesgue measure and denotes symmetric dif- ference △ A B := (A B) (B A). △ ∪ \ ∩ (a) Show that is a pseudometric on the set of bounded closed subsets. |∗ − ∗| (b) Show that is an -metric on the set of all open subsets. |∗ − ∗| ∞ C Completeness A metric space is called complete if every Cauchy sequence of points in convergesX in . X X 1.4. Exercise. Suppose that ρ is a positive continuous function on a complete metric space and ε> 0. Show that there is a point x such that X ∈ X ρ(x) < (1 + ε) ρ(y) · for any point y B(x, ρ(x)). ∈ Most of the time we will assume that a metric space is complete. The following construction produces a complete metric space ¯ for any given metric space . X X Completion. Given a metric space , consider the set of all Cauchy X C sequences in . Note that for any two Cauchy sequences (xn) and (yn) the right-handX side in ➊ is defined; moreover, it defines a pseudometric on C ➊ (xn) (yn) := lim xn yn . | − |C n→∞ | − |X The corresponding metric space ¯ is called completion of . Note that the original space X forms a dense subset inX its com- pletion ¯. More precisely, for eachX point x one can consider a X ∈ X constant sequence xn = x which is Cauchy. It defines a natural map 8 LECTURE 1. DEFINITIONS ¯. It is easy to check that this map is distance-preserving. In particular,X → X we can (and will) consider as a subset of ¯. X X 1.5. Exercise. Show that the completion of a metric space is com- plete. D Compact spaces Let us recall few equivalent definitions of compact metric spaces. 1.6. Definition. A metric space is compact if and only if one of the following equivalent conditionsK hold: (a) Every open cover of has a finite subcover. (b) For any open coverK of there is ε > 0 such that any ε-ball in lies in one elementK of the cover. (The value ε is called a LebesgueK number of the covering.) (c) Every sequence of points in has a subsequence that converges in . K (d) TheK space is complete and totally bounded; that is, for any ε> 0, the spaceK admits a finite cover by open ε-balls. K A subset N of a metric space is called ε-net if any point x lies on the distance less than ε fromK a point in N. Note that totally∈ K bounded spaces can be defined as spaces that admit a finite ε-net for any ε> 0. 1.7. Exercise. Show that a space is totally bounded if and only if it contains a compact ε-net for any Kε> 0. Let pack be the exact upper bound on the number of points ε X x ,...,xn such that xi xj > ε if i = j. 1 ∈ X | − | 6 If n = packε < , then the collection of points x1,...,xn is called a maximalXε-packing∞ . Note that n is the maximal number of open disjoint ε -balls in . 2 X 1.8. Exercise. Show that any maximal ε-packing is an ε-net. Conclude that a complete space is compact if and only if packε < < for any ε> 0. X X ∞ 1.9. Exercise. Let be a compact metric space and K f : K → K be a distance-nondecreasing map. Prove that f is an isometry; that is, f is a distance-preserving bijection. A metric space is called locally compact if any point in admits a compact neighborhood;X in other words, for any point x Xa closed ball B[x, r] is compact for some r> 0. ∈ X E. PROPER SPACES 9 E Proper spaces A metric space is called proper if all closed bounded sets in are compact. ThisX condition is equivalent to each of the followingX statements: For some (and therefore any) point p and any R< , the ⋄ ∈ X ∞ closed ball B[p, R]X is compact. The function distp : R is proper for some (and therefore ⋄ any) point p ;X that → is, for any compact set K R, its inverse image ∈ X ⊂ dist−1(K)= x : p x K p { ∈ X | − |X ∈ } is compact. 1.10. Exercise. Give an example of space which is locally compact but not proper. F Geodesics Let be a metric space and I a real interval.
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