A Course in Metric Geometry
Dmitri Burago
Yuri Burago
Sergei Ivanov
Department of Mathematics, Pennsylvania State University
Steklov Institute for Mathematics at St. Petersburg
Steklov Institute for Mathematics at St. Petersburg
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A Course in Metric Geometry
Dmitri Burago
Yuri Burago
Sergei Ivanov
Department of Mathematics, Pennsylvania State University
E-mail address: [email protected]
Steklov Institute for Mathematics at St. Petersburg
E-mail address: [email protected]
Steklov Institute for Mathematics at St. Petersburg
E-mail address: [email protected]
- Chapter 1. Metric Spaces Preface vii
- §1.1. Definitions
- §1.2. Examples
- §1.3. Metrics and Topology
- §1.4. Lipschitz Maps
- §1.5. Complete Spaces
- §1.6. Compact Spaces
- §1.7. Hausdorff Measure and Dimension
- Chapter 2. Length Spaces
- §2.1. Length Structures
- §2.2. First Examples of Length Structures
- §2.3. Length Structures Induced by Metrics
- §2.4. Characterization of Intrinsic Metrics
- §2.5. Shortest Paths
- §2.6. Length and Hausdorff Measure
- §2.7. Length and Lipschitz Speed
- Chapter 3. Constructions
- §3.1. Locality, Gluing and Maximal Metrics
- §3.2. Polyhedral Spaces
- §3.3. Isometries and Quotients
- §3.4. Local Isometries and Coverings iv Contents
- §3.5. Arcwise Isometries
- §3.6. Products and Cones
- Chapter 4. Spaces of Bounded Curvature
- §4.1. Definitions
- §4.2. Examples
- §4.3. Angles in Alexandrov Spaces and Equivalence of Definitions
- §4.4. Analysis of Distance Functions
- §4.5. The First Variation Formula
- §4.6. Nonzero Curvature Bounds and Globalization
- §4.7. Curvature of Cones
- Chapter 5. Smooth Length Structures
- §5.1. Riemannian Length Structures
- §5.2. Exponential Map
- §5.3. Hyperbolic Plane
- §5.4. Sub-Riemannian Metric Structures
- §5.5. Riemannian and Finsler Volumes
- §5.6. Besikovitch Inequality
- Chapter 6. Curvature of Riemannian Metrics
- §6.1. Motivation: Coordinate Computations
- §6.2. Covariant Derivative
- §6.3. Geodesic and Gaussian Curvatures
- §6.4. Geometric Meaning of Gaussian Curvature
- §6.5. Comparison Theorems
- Chapter 7. Space of Metric Spaces
- §7.1. Examples
- §7.2. Lipschitz Distance
- §7.3. Gromov–Hausdorff Distance
- §7.4. Gromov–Hausdorff Convergence
- §7.5. Convergence of Length Spaces
- Chapter 8. Large-scale Geometry
- §8.1. Noncompact Gromov–Hausdorff Limits
- §8.2. Tangent and Asymptotic Cones
- §8.3. Quasi-isometries Contents v
- §8.4. Gromov Hyperbolic Spaces
- §8.5. Periodic Metrics
- Chapter 9. Spaces of Curvature Bounded Above
- §9.1. Definitions and Local Properties
- §9.2. Hadamard Spaces
- §9.3. Fundamental Group of a Nonpositively Curved Space
- §9.4. Example: Semi-dispersing Billiards
- Chapter 10. Spaces of Curvature Bounded Below
- §10.1. One More Definition
- §10.2. Constructions and Examples
- §10.3. Toponogov’s Theorem
- §10.4. Curvature and Diameter
- §10.5. Splitting Theorem
- §10.6. Dimension and Volume
- §10.7. Gromov–Hausdorff Limits
- §10.8. Local Properties
- §10.9. Spaces of Directions and Tangent Cones
- §10.10. Further Information
- Bibliography
- Index
Preface
This book is not a research monograph or a reference book (although research interests of the authors influenced it a lot)—this is a textbook. Its structure is similar to that of a graduate course. A graduate course usually begins with a course description, and so do we.
Course description. The objective of this book is twofold. First of all, we wanted to give a detailed exposition of basic notions and techniques in the theory of length spaces, a theory which experienced a very fast development in the past few decades and penetrated into many other mathematical disci- plines (such as Group Theory, Dynamical Systems, and Partial Differential Equations). However, we have a wider goal of giving an elementary intro- duction into a broad variety of the most geometrical topics in geometry—the ones related to the notion of distance. This is the reason why we included metric introductions to Riemannian and hyperbolic geometries. This book tends to work with “easy-to-touch” mathematical objects by means of “easy- to-visualize” methods. There is a remarkable book [Gro3], which gives a vast panorama of “geometrical mathematics from a metric viewpoint”. Un- fortunately, Gromov’s book seems hardly accessible to graduate students and non-experts in geometry. One of the objectives of this book is to bridge the gap between students and researchers interested in metric geometry, and modern mathematical literature.
Prerequisite. It is minimal. We set a challenging goal of making the core part of the book accessible to first-year graduate students. Our expectations of the reader’s background gradually grow as we move further in the book. We tried to introduce and illustrate most of new concepts and methods by using their simplest case and avoiding technicalities that take attention
vii
viii Preface
away from the gist of the matter. For instance, our introduction to Riemann- ian geometry begins with metrics on planar regions, and we even avoid the notion of a manifold. Of course, manifolds do show up in more advanced sec- tions. Some exercises and remarks assume more mathematical background than the rest of our exposition; they are optional, and a reader unfamiliar with some notions can just ignore them. For instance, solid background in differential geometry of curves and surfaces in R^3 is not a mandatory prereq- uisite for this book. However, we would hope that the reader possesses some knowledge of differential geometry, and from time to time we draw analogies from or suggest exercises based on it. We also make a special emphasis on motivations and visualizations. A reader not interested in them will be able to skip certain sections. The first chapter is a clinic in metric topology; we recommend that the reader with a reasonable idea of metric spaces just skip it and use it for reference: it may be boring to read it. The last chapters are more advanced and dry than the first four.
Figures. There are several figures in the book, which are added just to make it look nicer. If we included all necessary figures, there would be at least five of them for each page.
- It is a must that the reader systematically studying this book makes a figure for every proposition, theorem, and construction!
Exercises. Exercises form a vital part of our exposition. This does not mean that the reader should solve all the exercises; it is very individual. The difficulty of exercises varies from trivial to rather tricky, and their importance goes all the way up from funny examples to statements that are 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 an exercise. You should try to prove each on your own, possibly after having a brief glance at our argument to get a hint. Just reading our proof is the last resort.
Optional material. Our exposition can be conditionally subdivided into two parts: core material and optional sections. Some sections and chapters are 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 optional sections (and even the less important parts of the core ones). Of course, this approach often requires going back and looking for important notions that were accidentally missed. A first reading can give a general picture of the theory, helping to separate its core and give a good idea of its logic. Then the reader goes through the book again, transforming theoretical knowledge into the genuine one by filling it with all the details, digressions, examples and experience that makes knowledge practical.
x Preface
situation. By now the scope of the theory of length spaces has grown quite far from its cradle (which was a theory of convex surfaces), including most of classical Riemannian geometry and many areas beyond it. At the same time, geometry of length spaces perhaps remains one of the most “hands- on” mathematical techniques. This combination of reasons urged us to write this “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. The authors’ research, which had essential impact on the book, was partially supported 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 write this book; S. Alexander, R. Bishop, and C. Croke for undertaking immense labor of thoroughly reading the manuscript—their numerous corrections, suggestions, and remarks were of invaluable help; S. Buyalo for many useful comments and suggestions for Chapter 9; K. Shemyak for preparing most of the figures; and finally a group of graduate students at Penn State who took 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 twice as many of them are still left for the reader).
Chapter 1
Metric Spaces
The purpose of the major part of the chapter is to set up notation and to refresh the reader’s knowledge of metric spaces and related topics in point- set topology. Section 1.7 contains minimal information about Hausdorff measure and dimension.
It may be a good idea to skip this chapter and use it only for reference, or to look through it briefly to make sure that all examples are clear and exercises are obvious.
1.1. Definitions
Definition 1.1.1. Let X be an arbitrary set. A function d : X × X → R ∪ {∞} is a metric on X if the following conditions are satisfied for all x, y, z ∈ X.
(1) Positiveness: d(x, y) > 0 if x 6 = y, and d(x, x) = 0. (2) Symmetry: d(x, y) = d(y, x). (3) Triangle inequality: d(x, z) ≤ d(x, y) + d(y, z).
A metric space is a set with a metric on it. In a formal language, a metric space is a pair (X, d) where d is a metric on X. Elements of X are called points of the metric space; d(x, y) is referred to as the distance between points x and y.
When the metric in question is clear from the context, we also denote the distance between x and y by |xy|.
Unless different metrics on the same set X are considered, we will omit an explicit reference to the metric and write “a metric space X” instead of “a metric space (X, d).”
1.2. Examples 3
1.2. Examples
Various examples of metric spaces will appear everywhere in the course. In this section we only describe several important ones to begin with. For many of them, verification of the properties from Definition 1.1.1 is trivial and is left for the reader.
Example 1.2.1. One can define a metric on an arbitrary set X by
|xy| =
0 if x = y, 1 if x 6 = y.
This example is not particularly interesting but it can serve as the initial point for many constructions.
Example 1.2.2. The real line, R, is canonically equipped with the distance |xy| = |x − y|, and thus can be considered as a metric space. There is an immense variety of other metrics on R; for instance, consider dlog(x, y) = log |x − y|.
Example 1.2.3. The Euclidean plane, R^2 , with its standard distance, is another familiar metric space. The distance can be expressed by the Pythagorean formula,
|xy| = |x − y| =
(x 1 − y 1 )^2 + (x 2 − y 2 )^2
where (x 1 , x 2 ) and (y 1 , y 2 ) are coordinates of points x and y. The triangle inequality for this metric is known from elementary Euclidean geometry. Alternatively, it can be derived from the Cauchy inequality.
Example 1.2.4 (direct products). Let X and Y be two metric spaces. We define a metric on their direct product X × Y by the formula
|(x 1 , y 1 )(x 2 , y 2 )| =
|x 1 x 2 |^2 + |y 1 y 2 |^2.
In particular, R × R = R^2.
Exercise 1.2.5. Derive the triangle inequality for direct products from the triangle inequality on the Euclidean plane.
Example 1.2.6. Recall that the coordinate n-space Rn^ is the vector space of all n-tuples (x 1 ,... , xn) of real numbers, with component-wise addition and multiplication by scalars. It is naturally identified with the multiple direct product R × · · · × R (n times). This defines the standard Euclidean distance,
|xy| =
(x 1 − y 1 )^2 + · · · + (xn − yn)^2
where x = (x 1 ,... , xn) and y = (y 1 ,... , yn).
4 1. Metric Spaces
Example 1.2.7 (dilated spaces). This simple construction is similar to obtaining one set from another by means of a homothety map. Let X be a metric space and λ > 0. The metric space λX is the same set X equipped with another distance function dλX which is defined by dλX (x, y) = dX (x, y) for all x, y ∈ X, where dX is the distance in X. The space λX is referred to as X dilated (or rescaled) by λ.
Example 1.2.8 (subspaces). If X is a metric space and Y is a subset of X, then a metric on Y can be obtained by simply restricting the metric from X. In other words, the distance between points of Y is equal to the distance between the same points in X.
Restricting the distance is the simplest but not the only way to define a metric on a subset. In many cases it is more natural to consider an intrinsic metric, which is generally not equal to the one restricted from the ambient space. The notion of intrinsic metric will be explained further in the course, but its intuitive meaning can be illustrated by the following example of the intrinsic metric on a circle.
Example 1.2.9. The unit circle, S^1 , is the set of points in the plane lying at distance 1 from the origin. Being a subset of the plane, the circle carries the restricted Euclidean metric on it. We define an alternative metric by setting the distance between two points as the length of the shorter arc between them. For example, the arc-length distance between two opposite points of the circle is equal to π. The distance between adjacent vertices of a regular n-gon (inscribed into the circle) is equal to 2π/n.
Exercise 1.2.10. (a) Prove that any circle arc of length less or equal to π, equipped with the above metric, is isometric to a straight line segment.
(b) Prove that the entire circle with this metric is not isometric to any subset of the plane (regarded with the restriction of Euclidean distance onto this subset).
1.2.1. Normed vector spaces.
Definition 1.2.11. Let V be a vector space^1. A function | · | : V → R is a norm on V if the following conditions are satisfied for all v, w ∈ V and k ∈ R.
(1) Positiveness: |v| > 0 if v 6 = 0, and | 0 | = 0. (2) Positive homogeneity: |kv| = |k||v|. (3) Subadditivity (triangle inequality): |v + w| ≤ |v| + |w|.
(^1) All normed spaces here are ones over R.
6 1. Metric Spaces
Definition 1.2.17. A scalar product is a symmetric bilinear form F whose associated quadratic form is positive definite, i.e., F (x, x) > 0 for all x 6 = 0. A Euclidean space is a vector space with a scalar product on it.
We will use notation 〈·, ·〉 for various scalar products.
Definition 1.2.18. A norm associated with a scalar product 〈·, ·〉 is defined by the formula |v| =
〈v, v〉. A norm is called Euclidean if it is associated with some scalar product.
For example, the standard norm in Rn^ is associated with the scalar product defined by 〈x, y〉 =
xiyi where x = (x 1 ,... , xn) and y = (y 1 ,... , yn).
Exercise 1.2.19. Prove the triangle inequality for a norm associated with a scalar product.
Hint: First, reduce the triangle inequality to: 〈v, w〉 ≤ |v| · |w| for any two vectors v and w. Then expand the relation 〈v − tw, v − tw〉 ≥ 0 and substitute t = 〈v, v〉 / 〈w, w〉. Another way to prove the triangle inequality is to combine Proposition 1.2.22 and the triangle inequality for Rn.
Since a scalar product is uniquely determined by its associated norm, a Euclidean space could be defined as a normed space whose norm is Euclid- ean. The following exercise give an explicit characterization of Euclidean spaces among the normed spaces.
Exercise 1.2.20. Prove that a norm | · | on a vector space V is Euclidean if and only if |v + w|^2 + |v − w|^2 = 2(|v|^2 + |w|^2 )
for all v, w ∈ V.
Exercise 1.2.21. Show that Rn 1 and Rn ∞ are not Euclidean spaces for n > 1.
Two vectors in a Euclidean space are called orthogonal if their scalar product is zero. An orthonormal frame is a collection of mutually orthogonal unit vectors. Vectors of an orthonormal frame are linearly independent (prove this!). An orthonormal 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 ortho- normal basis. Let dim V = n and {e 1 ,... , en} be such a basis. Every vector x ∈ V can be uniquely represented as a linear combination
xiei for some xi ∈ R. Since all scalar products of vectors ei are known, we can find the scalar product of any linear combination, namely 〈∑ xiei,
yiei
xiyi.
1.3. Metrics and Topology 7
This implies the following
Proposition 1.2.22. Every n-dimensional Euclidean space is isomorphic to Rn. This means that there is a linear isomorphism f : Rn^ → V such that 〈f (x), f (y)〉 = 〈x, y〉 for all x, y ∈ Rn. In particular, these spaces are isometric.
Proof. Define f ((x 1 ,... , xn)) =
xiei where {ei} 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 R^2 , 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 distance-preserving map from one Euclid- ean space to another is an affine 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 f : V → V such that f (v) = w.
1.2.3. Spheres.
Example 1.2.25. The n-sphere Sn^ is the set of unit vectors in Rn+1, i.e., Sn^ = {x ∈ Rn+1^ : |x| = 1}. The angular metric on Sn^ is defined by
d(x, y) = arccos 〈x, y〉.
In other words, the spherical distance is defined as the Euclidean angle between unit vectors. It equals the length of the shorter arc of a great circle connecting x and y in the sphere. Another formula for this metric is
d(x, y) = 2 arcsin
|x − y| 2
The metric on the circle described in Example 1.2.9 is a partial case of this example.
1.3. Metrics and Topology
Definition 1.3.1. Let X be a metric space, x ∈ X and r > 0. The set formed by the points at distance less than r from x is called an (open metric) ball of radius r centered at x. We denote this ball by Br(x). Similarly, a closed ball Br(x) is the set of points whose distances from x are less than or equal to r.
1.4. Lipschitz Maps 9
1.4. Lipschitz Maps
Definition 1.4.1. Let X and Y be metric spaces. A map f : X → Y is called Lipschitz if there exists a C ≥ 0 such that |f (x 1 )f (x 2 )| ≤ C|x 1 x 2 | for all x 1 , x 2 ∈ X. Any suitable value of C is referred to as a Lipschitz constant of f. The minimal Lipschitz constant is called the dilatation of f and denoted by dil f. The dilatation of a non-Lipschitz function is infinity.
A map with Lipschitz constant 1 is called nonexpanding.
Exercise 1.4.2. The distance from a point x to a set S in a metric space is defined by dist(x, S) = infy∈S |xy|. Prove that dist(·, S) is a nonexpanding function.
Proposition 1.4.3. (1) All Lipschitz maps are continuous.
(2) If f : X → Y and g : Y → Z are Lipschitz maps, then g ◦ f is Lipschitz and dil(g ◦ f ) ≤ dil f · dil g. (3) The set of real-valued Lipschitz functions on a metric space (and, more generally, the set of Lipschitz functions from a metric space to a normed space) is a vector space. One has dil(f +g) ≤ dil f +dil g, dil(λf ) = |λ| dil f for any Lipschitz functions f and g and λ ∈ R.
Definition 1.4.4. Let X and Y be metric spaces. A map f : X → Y is called locally Lipschitz if every point x ∈ X has a neighborhood U such that f |U is Lipschitz. The dilatation of f at x is defined by
dilx f = inf{dil f |U : U is a neighborhood of x}.
Exercise 1.4.5. Let X be a metric space. Prove that dil f = supx∈R dilx f for any map f : R → X. Prove the same statement with R replaced by S^1 with the metric described in Exercise 1.2.9. Show that it is not true for S^1 with the metric restricted from R^2.
Definition 1.4.6. Let X and Y be metric spaces. A map f : X → Y is called bi-Lipschitz if there are positive constants c and C such that
c|x 1 x 2 | ≤ |f (x 1 )f (x 2 )| ≤ C|x 1 x 2 |
for all x 1 , x 2 ∈ X.
Clearly every bi-Lipschitz map is a homeomorphism onto its image.
Definition 1.4.7. Two metrics d 1 and d 2 on the same set X are called Lipschitz equivalent if there are positive constants c and C such that
c · d 1 (x, y) ≤ d 2 (x, y) ≤ C · d 1 (x, y)
for all x, y ∈ X.
10 1. Metric Spaces
In other words, d 1 and d 2 are Lipschitz equivalent if the identity is a bi- Lipschitz map from (X, d 1 ) to (X, d 2 ). Clearly this is an equivalence relation on the set of metrics in X. Lipschitz equivalent metrics determine the same topology.
Exercise 1.4.8. Give an example of two metrics on the same set that determine the same topology but are not Lipschitz equivalent.
Exercise 1.4.9. Let X and Y be metric spaces. Prove that the following three metrics on X × Y are Lipschitz equivalent:
- The metric defined in Example 1.2.4.
- d 1 ((x 1 , y 1 ), (x 2 , y 2 )) = |x 1 x 2 | + |y 1 y 2 |.
- d∞((x 1 , y 1 ), (x 2 , y 2 )) = max{|x 1 x 2 |, |y 1 y 2 |}.
Exercise 1.4.10. Let X be a metric space. Prove that its metric is a Lipschitz function on X × X where X × X is regarded to have any of the metrics from the previous exercise.
We conclude this section with the following important theorem about normed spaces.
Theorem 1.4.11. 1. Two norms on a vector space determine the same topology if and only if they are Lipschitz equivalent;
- All norms on a finite-dimensional vector space are Lipschitz equiva- lent.
1.5. Complete Spaces
Definition 1.5.1. A sequence {xn} in a metric space is called a Cauchy sequence if |xnxm| → 0 as n, m → ∞. The precise meaning of this is the following: for any ε > 0 there exists an n 0 such that |xnxm| < ε whenever n ≥ n 0 and m ≥ n 0.
A metric space is called complete if every Cauchy sequence in it has a limit.
It is known from analysis (see e.g. [Mun]) that R is a complete space. It easily follows that Rn^ is complete for all n. R \ { 0 } is an example of a noncomplete space; a sequence that would converge to zero in R is a Cauchy sequence that has no limit in this space. (Note that a converging sequence is always a Cauchy one.)
Exercise 1.5.2. Prove that completeness is preserved by a bi-Lipschitz homeomorphism. In particular, Lipschitz equivalent metrics share complete- ness or noncompleteness.