**A** **Course** **in** **Metric** GeometryDmitri BuragoYuri BuragoSergei IvanovDepartment of Mathematics, Pennsylvania State UniversityE-mail address: burago@math.psu.eduSteklov Institute for Mathematics at St. PetersburgE-mail address: burago@pdmi.ras.ruSteklov Institute for Mathematics at St. PetersburgE-mail address: svivanov@pdmi.ras.ruContentsPreface viiChapter 1.

**Metric** Spaces 1§1.1. Deﬁnitions 1§1.2. Examples 3§1.3. Metrics and Topology 7§1.4. Lipschitz Maps 9§1.5. Complete Spaces 10§1.6. Compact Spaces 13§1.7. Hausdorﬀ Measure and Dimension 17Chapter 2. Length Spaces 25§2.1. Length Structures 25§2.2. First Examples of Length Structures 30§2.3. Length Structures Induced by Metrics 33§2.4. Characterization of Intrinsic Metrics 38§2.5. Shortest Paths 44§2.6. Length and Hausdorﬀ Measure 53§2.7. Length and Lipschitz Speed 55Chapter 3. Constructions 59§3.1. Locality, Gluing and Maximal Metrics 59§3.2. Polyhedral Spaces 67§3.3. Isometries and Quotients 74iiiiv Contents§3.4. Local Isometries and Coverings 78§3.5. Arcwise Isometries 85§3.6. Products and Cones 87Chapter 4. Spaces of Bounded Curvature 101§4.1. Deﬁnitions 101§4.2. Examples 109§4.3. Angles

**in** Alexandrov Spaces and Equivalence of Deﬁnitions 114§4.4. Analysis of Distance Functions 119§4.5. The First Variation Formula 121§4.6. Nonzero Curvature Bounds and Globalization 126§4.7. Curvature of Cones 131Chapter 5. Smooth Length Structures 135§5.1. Riemannian Length Structures 136§5.2. Exponential Map 150§5.3. Hyperbolic Plane 154§5.4. Sub-Riemannian

**Metric** Structures 178§5.5. Riemannian and Finsler Volumes 193§5.6. Besikovitch Inequality 202Chapter 6. Curvature of Riemannian Metrics 209§6.1. Motivation: Coordinate Computations 211§6.2. Covariant Derivative 214§6.3. Geodesic and Gaussian Curvatures 221§6.4. Geometric Meaning of Gaussian Curvature 226§6.5. Comparison Theorems 237Chapter 7. Space of

**Metric** Spaces 241§7.1. Examples 242§7.2. Lipschitz Distance 249§7.3. Gromov–Hausdorﬀ Distance 251§7.4. Gromov–Hausdorﬀ Convergence 260§7.5. Convergence of Length Spaces 265Chapter 8. Large-scale

**Geometry** 271§8.1. Noncompact Gromov–Hausdorﬀ Limits 271§8.2. Tangent and Asymptotic Cones 275Contents v§8.3. Quasi-isometries 277§8.4. Gromov Hyperbolic Spaces 284§8.5. Periodic Metrics 298Chapter 9. Spaces of Curvature Bounded Above 307§9.1. Deﬁnitions and Local Properties 308§9.2. Hadamard Spaces 324§9.3. Fundamental Group of

**a** Nonpositively Curved Space 338§9.4. Example: Semi-dispersing Billiards 341Chapter 10. Spaces of Curvature Bounded Below 351§10.1. One More Deﬁnition 352§10.2. Constructions and Examples 354§10.3. Toponogov’s Theorem 360§10.4. Curvature and Diameter 364§10.5. Splitting Theorem 366§10.6. Dimension and Volume 369§10.7. Gromov–Hausdorﬀ Limits 376§10.8. Local Properties 378§10.9. Spaces of Directions and Tangent Cones 390§10.10. Further Information 398Bibliography 405Index 409PrefaceThis book is not

**a** research monograph or

**a** reference book (althoughresearch interests of the authors inﬂuenced it

**a** lot)—this is

**a** textbook.Its structure is similar to that of

**a** graduate course.

**A** graduate courseusually begins with

**a** **course** description, and so do we.Course description. The objective of this book is twofold. First of all, wewanted to give

**a** detailed exposition of basic notions and techniques

**in** thetheory of length spaces,

**a** theory which experienced

**a** very fast developmentin the past few decades and penetrated into many other mathematical disci-plines (such as Group Theory, Dynamical Systems, and Partial DiﬀerentialEquations). However, we have

**a** wider goal of giving an elementary intro-duction into

**a** broad variety of the most geometrical topics

**in** geometry—theones related to the notion of distance. This is the reason why we includedmetric introductions to Riemannian and hyperbolic geometries. This booktends to work with “easy-to-touch” mathematical objects by means of “easy-to-visualize” methods. There is

**a** remarkable book [Gro3], which gives avast panorama of “geometrical mathematics from

**a** **metric** viewpoint”. Un-fortunately, Gromov’s book seems hardly accessible to graduate studentsand non-experts

**in** geometry. One of the objectives of this book is to bridgethe gap between students and researchers interested

**in** **metric** geometry, andmodern mathematical literature.Prerequisite. It is minimal. We set

**a** challenging goal of making the corepart of the book accessible to ﬁrst-year graduate students. Our expectationsof the reader’s background gradually grow as we move further

**in** the book.We tried to introduce and illustrate most of new concepts and methodsby using their simplest case and avoiding technicalities that take attentionviiviii Prefaceaway from the gist of the matter. For instance, our introduction to Riemann-ian

**geometry** begins with metrics on planar regions, and we even avoid thenotion of

**a** manifold. Of course, manifolds do show up

**in** more advanced sec-tions. Some exercises and remarks assume more mathematical backgroundthan the rest of our exposition; they are optional, and

**a** reader unfamiliarwith some notions can just ignore them. For instance, solid background indiﬀerential

**geometry** of curves and surfaces

**in** R3is not

**a** mandatory prereq-uisite for this book. However, we would hope that the reader possesses someknowledge of diﬀerential geometry, and from time to time we draw analogiesfrom or suggest exercises based on it. We also make

**a** sp ecial emphasis onmotivations and visualizations.

**A** reader not interested

**in** them will be ableto skip certain sections. The ﬁrst chapter is

**a** clinic

**in** **metric** topology; werecommend that the reader with

**a** reasonable idea of

**metric** spaces just skipit and use it for reference: it may be boring to read it. The last chaptersare more advanced and dry than the ﬁrst four.Figures. There are several ﬁgures

**in** the book, which are added just tomake it look nicer. If we included all necessary ﬁgures, there would be atleast ﬁve of them for each page.• It is

**a** must that the reader systematically studying this book makesa ﬁgure for every proposition, theorem, and construction!Exercises. Exercises form

**a** vital part of our exposition. This does notmean that the reader should solve all the exercises; it is very individual.The diﬃculty of exercises varies from trivial to rather tricky, and theirimportance goes all the way up from funny examples to statements thatare extensively used later

**in** the book. This is often indicated

**in** the text.It is

**a** very helpful strategy to perceive every proposition and theorem as anexercise. You should try to prove each on your own, possibly after havinga brief glance at our argument to get

**a** hint. Just reading our proof is thelast resort.Optional material. Our exposition can be conditionally subdivided intotwo parts: core material and optional sections. Some sections and chaptersare preceded by

**a** brief plan, which can be used as

**a** guide through them.It is usually

**a** good idea to begin with

**a** ﬁrst reading, skipping all optionalsections (and even the less important parts of the core ones). Of course, thisapproach often requires going back and looking for important notions thatwere accidentally missed.

**A** ﬁrst reading can give

**a** general picture of thetheory, helping to separate its core and give

**a** good idea of its logic. Thenthe reader goes through the book again, transforming theoretical knowledgeinto the genuine one by ﬁlling it with all the details, digressions, examplesand experience that makes knowledge practical.Preface ixAbout

**metric** geometry. Whereas the borderlines between mathemati-cal disciplines are very conditional,

**geometry** historically began from very“down-to-earth” notions (even literally). However, for most of the last cen-tury it was

**a** common belief that “geometry of manifolds” basically boileddown to “analysis on manifolds”. Geometric methods heavily relied on dif-ferential machinery, as it can be guessed even from the name “Diﬀerentialgeometry”. It is now understood that

**a** tremendous part of

**geometry** es-sentially belongs to

**metric** geometry, and the diﬀerential apparatus can beused just to deﬁne some class of objects and extract the starting data tofeed into the synthetic methods. This certainly cannot be applied to allgeometric notions. Even the curvature tensor remains an obscure monster,and the geometric meaning of only some of its simplest appearances (suchas the sectional curvature) are more or less understood. Many modern re-sults involving more advanced structures still sound quite analytical. Onthe other hand, expelling analytical machinery from

**a** certain sphere ofdeﬁnitions and arguments brought several major beneﬁts. First of all, itenhanced mathematical understanding of classical objects (such as smoothRiemannian manifolds) both ideologically, and by concrete results. From amethodological viewpoint, it is important to understand what assumptions aparticular result relies on; for instance,

**in** this respect it is more satisfying toknow that geometrical properties of positively curved manifolds are basedon

**a** certain inequality on distances between quadruples of points ratherthan on some properties of the curvature tensor. This is very similar totwo ways of thinking about convex functions. One can say that

**a** functionis convex if its second derivative is nonnegative (notice that the deﬁnitionalready assumes that the function is smooth, leaving out such functions asf(x) = |x|). An alternative deﬁnition says that

**a** function is convex if itsepigraph (the set {(x, y) : y ≥ f(x)}) is; the latter deﬁnition is equivalentto Jensen’s inequality f(αx + βy) ≤ αf(x) + βf(y) for all nonnegative α, βwith α + β = 1, and it is robust and does not rely on the notion of

**a** limit.From this viewpoint, the condition f≥ 0 can be regarded as

**a** convenientcriterion for

**a** smooth function to be convex.As

**a** more speciﬁc illustration of an advantage of this way of thinking,imagine that one wants to estimate

**a** certain quantity over all metricson

**a** sphere. It is so tempting to study

**a** **metric** for which the quantityattains its maximum, but alas this

**metric** may fail exist within smoothmetrics, or even metrics that induce the same topology. It turns out thatit still may exist if we widen our search to

**a** class of more general lengthspaces. Furthermore, mathematical topics whose study used to lie outsidethe range of noticeable applications of geometrical technique now turnedout to be traditional objects of methods originally rooted

**in** diﬀerentialgeometry. Combinatorial group theory can serve as

**a** model example of thisx Prefacesituation. By now the scope of the theory of length spaces has grown quitefar from its cradle (which was

**a** theory of convex surfaces), including mostof classical Riemannian

**geometry** and many areas beyond it. At the sametime,

**geometry** of length spaces perhaps remains one of the most “hands-on” mathematical techniques. This combination of reasons urged us to writethis “beginners’

**course** **in** **geometry** from

**a** length structure viewpoint”.Acknowledgements. The authors enjoyed hospitality and excellent work-ing conditions during their stays at various institutions, including the Uni-versity of Strasbourg, ETH Zurich, and Cambridge University. These un-forgettable visits were of tremendous help to the progress of this book. Theauthors’ research, which had essential impact on the book, was partiallysupported by the NSF Foundation, the Sloan Research Fellowship, CRDF,RFBR, and Shapiro Fund at Penn State, whose help we gratefully acknowl-edge. The authors are grateful to many people for their help and encour-agement. We want to especially thank M. Gromov for provoking us to writethis book; S. Alexander, R. Bishop, and C. Croke for undertaking immenselabor of thoroughly reading the manuscript—their numerous corrections,suggestions, and remarks were of invaluable help; S. Buyalo for many usefulcomments and suggestions for Chapter 9; K. Shemyak for preparing mostof the ﬁgures; and ﬁnally

**a** group of graduate students at Penn State whotook

**a** Math 597c

**course** using our manuscript as the base text and cor-rected dozens of typos and small errors (though we are conﬁdent that twiceas many of them are still left for the reader).[...]... automatically satisﬁed From now on we reserve the word path for maps of intervals:

**a** path γ

in **a** (topological) space X is

**a** (continuous) map γ : I → X deﬁned on an interval I ⊂ R By an interval we mean any connected subset of the real line; it may be open or closed, ﬁnite or

**in** nite, and

**a** single point is counted as an interval Since

**a** path is

**a** map one can speak about its image, restrictions, etc

**A** length... such covers Speaking about “the total measure of

**a** cover” one means here that certain measure is already assigned to simple sets To deﬁne Hausdorﬀ n-dimensional measure on

**a** **metric** space, one could proceed along the same lines: cover

**a** set by

**metric** balls such that all their radii are less than ε For each ball, consider

**a** Euclidean ball of the same radius and add their volumes for all balls from the... is an orthonormal basis This proposition allows one to apply elementary Euclidean

**geometry** to general Euclidean spaces For example, since any two-dimensional subspace of

**a** Euclidean space is isomorphic to R2 , any statement involving only two vectors and their linear combinations can be automatically transferred from the standard Euclidean plane to all Euclidean spaces Exercise 1.2.23 Prove that any... speaking,

**a** length structure consists of

**a** class of admissible paths for which we can measure their length, and the length itself, which is

**a** correspondence assigning

**a** nonnegative number to every path from the class Both the class and the correspondence have to possess several natural properties;

**in** all reasonable examples (and

**in** particular

**in** all examples

**in** this book) these requirements are automatically... nonnegative Exercise 1.7.4 Let µ be

**a** measure Prove the following statements: (a) Let {Ai }∞ be

**a** sequence of measurable sets such that Ai ⊂ Ai+1 for i=1 all i Then the sequence {µ(Ai )} is nondecreasing and lim µ(Ai ) = µ( Ai ) (b) Let {Ai }∞ be

**a** sequence of measurable sets such that Ai ⊃ Ai+1 i=1 for all i, and assume that µ (A1 ) < ∞ Then the sequence {µ(Ai )} is nonincreasing and lim µ(Ai ) = µ( Ai... that any distance-preserving map from one Euclidean space to another is an

**a** ne map, that is,

**a** composition of

**a** linear map and

**a** parallel translation Show by example that this is generally not true for arbitrary normed spaces Exercise 1.2.24 Let V be

**a** ﬁnite-dimensional normed space Prove that V is Euclidean if and only if for any two vectors v, w ∈ V such that |v| = |w| there exists

**a** linear isometry... that (a) Any subset of

**a** totally bounded set is totally bounded (b)

**In** Rn , any bounded set (that is,

**a** set whose diameter is ﬁnite) is totally bounded Exercise 1.6.4

**A** set S

in **a** metric space is called ε-separated, for an ε > 0, if |xy| ≥ ε for any two diﬀerent points x, y ∈ S Prove that 1 If there exists an (ε/3)-net of cardinality n, then an ε-separated set cannot contain more than n points 2

**A** maximal... frame can be obtained from any collection of linearly independent vectors by

**a** standard Gram–Schmidt orthogonalization procedure

**In** particular,

**a** ﬁnite-dimensional Euclidean space V possesses an orthonormal basis Let dim V = n and {e1 , , en } be such

**a** basis Every vector x ∈ V can be uniquely represented as

**a** linear combination xi ei for some xi ∈ R Since all scalar products of vectors ei are known,... 1.7.8 is actually

**a** measure Thus we obtain Theorem 1.7.9 For any

**metric** space X and any d ≥ 0, µd is

**a** measure on the Borel σ-algebra of X Exercise 1.7.10 Prove that 0-dimensional Hausdorﬀ measure of

**a** set is its cardinality

**In** other words, µ0 (X) is

**a** number of points

**in** X if X is

**a** ﬁnite set, and µ0 (X) = ∞ if X is an

**in** nite set Exercise 1.7.11 Let X and Y be

**metric** spaces and f : X → Y

**a** locally Lipschitz... Earth (like us) have to take long detours with lots of ups and downs; see Figure 2.1

**A** B PSfrag replacements

**A** B Figure 2.1:

**A** crow ﬂies” along the segment AB; for

**a** pedestrian it probably takes longer 25 26 2 Length Spaces This little philosophical digression contains

**a** very clear mathematical moral:

**in** many cases, we have to begin with length of paths as the primary notion and only after that can . convex surfaces), including mostof classical Riemannian geometry and many areas beyond it. At the sametime, geometry of length spaces perhaps remains one. metrics,namely a metric space with possibly in nite distances splits canonically intosubspaces that carry ﬁnite metrics and are separated from one another by in nite