SURFACES OF FINITE CURVATURE

1. Introduction

1.1. Gauss-Bonnet, classical. One of the deepest and most impor-tant theorems in classical differential geometry of surfaces is the the-orem of Gauss–Bonnet. It has a local and a global form and can beformulated for general curves and more geometrically for polygons ortriangles.

We recall the local form for triangles. Let U be a smooth surfacewith a Riemannian metric homeomorphic to the 2-dimensional plane.Classically U is an immersed disc in R3 and the Riemannian metric isinduced by the so called first fundamental form.

There are well-defined notions of lengths of curves and of shortestcurves between points. These shortest curves are smooth solutions ofthe geodesic equation, the so-called geodesics. Given a simple triangle∆ in U , thus simple closed curve built by a concatenation of 3 geodesics,we have well-defined vertices and angles α1, α2, α3 at these vertices.

The theorem of Gauss-Bonnet now tells us that the sum of the anglesin the triangle ∆ can be computed from the so-called Gaussian curva-ture, known also as sectional curvature in Riemannian geometry. Thecurvature (in our case of surfaces) is a real number κ(x) assigned toeach point x ∈ U , which is a measure of how non-Euclidean the metricaround this point behaves infinitesimally (in a precise and non-trivialsense).

Denote by δ(∆) the defect of the triangle ∆ defined as

δ(∆) :=∑

αi − π .

The theorem of Gauss–Bonnet states that

δ(∆) =

∫∆

κ(x) dvol(x) ,

where vol denotes the natural area measure on U .Turning to the global version of the theorem of Gauss–Bonnet assume

now that M is a compact two-dimensional surface with a Riemannianmetric. Again the notions of shortest curves, geodesics, angles betweenthem and curvature are well-defined and M is equipped with its natural

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area measure vol. One can triangulate M into simple geodesic trian-gles, thus one can dissect M into simple triangles, each pair of themintersecting in the empty set, a common side or a common vertex.

Summing the right-hand sides in the local theorem of Gauss-Bonnetone obtains the integral of the curvature∫

M

κ(x) dvol(x)

Summing the left-hand sides we get

2π · V − π · F ,where V and F are the numbers of vertices and triangles in the trian-gulation. In the triangulation, the number F of triangles satisfies

3F = 2E ,

where E is the number of edges. Thus, we arrive at the global formulaof Gauss–Bonnet

2πχ(M) =

∫M

κ(x) dvol(x) ,

with

χ(M) = V − E + F

the Euler characteristic of M .This equality has many different consequences. The first (usually

proven on the way to this formula) is the statement that the Eulercharacteristic V − E + F does not depend on the triangulation. It isa topological invariant of M . In fact it can also be computed by thesame formula using decomposition into polygons different from trian-gles. This invariant equals to 2 on the sphere, 0 for the torus and −2kfor the torus with k handles attached.

Secondly, it proves the highly non-trivial statement, that the inte-gral of the curvature over M does not depend on the choice of theRiemannian metric.

1.2. Gauss-Bonnet, simplicial. Now we slightly change the subjectand consider a simplical surface. For simplicity we consider the bound-ary M of a (possibly non-convex) polyhedron in R3. Subdividing all ofits faces we get a triangulation of M . Now, all triangles involved areEuclidean and the sum of all its defects is 0.

However, we can express this 0 in another way, again first summingup the angles around the vertices. We get

π · F =∑

(αv) ,2

where αv denotes the sum of all angles at the vertex v and the sum-mation is taken over all vertices v in the triangulation.

Write κ(v) = 2π − αv and call it the curvature of the simplicialsurface M at the vertex v, we see∑

v

κ(v) = 2π · V − π · F = 2πχ(M) .

Thus, the formula of Gauss–Bonnet holds true, if we replace thedifferential-geometric measure κ(x) · vol(x) by a discrete measure con-centrated in the vetrices of M and measuring the difference of the totalangle at the vertices from 2π.

1.3. Aim and plan. The central aim of this lecture will be to under-stand the connection between the theorems of Gauss–Bonnet in thediscrete and in the classical setting and extend it to a general theory ofsurfaces for which a version of the theorem holds true for some curva-ture measure, generalizing the classical curvature times area measureand the discrete measure concentrated in the vertices in the simplicialcase.

On the way, we will learn some metric geometry and repeat (orshortly sketch) results in classical differential geometry of surfaces.

We will start with learning some basics about shortest curves andangles.

Since we will work on surfaces almost exclusively, we will need somebasics in classical two-dimensional topology, related to Jordan’s curvetheorem. We will recall that as well.

We will learn and discuss some important metric constructions likegluing and coning, describe simplicial metric spaces and understandthe validity also of the local version of the theorem of Gauss-Bonnet insuch spaces.

After these preparations we will turn to surfaces on which a theoremof Gauss–Bonnet holds true and will try to understand some aspects oftheir theory. This theory has been developed by Alexandrov, Reshet-nyak and Zallgaller and developped further by students of Alexandrov.

The plan is to follow the survey of Y. Reshetnyak, [Res93]. At the be-ginning we will recall some basic material in metric geometry, [BBI01]can be used as a good source.

2. Length spaces

2.1. Curves. Let (X, d) be a metric space. A curve in X is a contin-uous map γ : I → X of an interval I ⊂ R.

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A curve γ : I → X is a reparametrization of another curve γ : J → Xif there exist a non-decreasing surjection s : J → I such that γ = γ s.

A curve is called simple if it injective. It is called closed if I = [a, b]and γ(a) = γ(b). It is called simple closed if I = [a, b], γ is injective on[a, b) and γ(a) = γ(b).

We will often (in particular, if γ is simple or simple closed) identifya curve and its image in X by abuse of notations.

2.2. Length of curves. For a curve γ : I → X the length of γ is

`(γ) = `X(γ) = `d(γ) = sup∑i

d(γ(ti), γ(ti+1)) ,

where the supremum is taken over all sequences t0 ≤ t1 ≤ ... ≤ tn ∈ I.The following properties hold:1) If X = Rn with its Euclidean metric and γ is C1 then

`(γ) =

∫I

|γ′(t)| dt .

The same formula holds for C1 curves in Riemannian manifolds M .2) `(γ) ≥ d(γ(a), γ(b)), for all a, b ∈ I. The curve γ is constant if

and only if `(γ) = 0.3) If I is subdivided in two intervals I = I− ∪ I+ intersecting at

one point, thus γ is represented as a concatenation of γ1 = γ|I− andγ2 = γ|I+ , then

`(γ) = `(γ1) + `(γ2) .

4) Length does not change under reparametrizations and change oforientation.

5) If γ is L-Lipschitz, then `(γ) ≤ L · H1(I).Here and later we denote by Hn (for us only n = 1, 2 are relevant)

the n-dimensional Hausdorff measure. On Rn the measureHn coincideswith the Lebesgue measure.

6) We say that γ is parametrized by arclength if, for any subintervalJ ⊂ I we have `(γ|J) = H1(J).

Any curve parametrized by arclength is 1-Lipschitz.A C1-curve γ : I → Rn (or in a Riemannian manifold M) is parama-

trized by arclength if and only if |γ′(t)| = 1 for all t ∈ I.Every curve of finite length (called a rectifiable curve) admits a

parametrization by arclength. The same is true for locally rectifiablecurves, thus curves, whose restrictions to any compact subinterval of Ihave finite length.

7) If curves γn : I → X converge pointwise to γ : I → X then

`(γ) ≤ lim inf `(γi) .4

The theorem of Arzela-Ascoli implies that any sequence of Lipschitzcurves with the same Lipschitz constant whose images are containedin a compact set always has a convergent subsequence. Together withthe above semicontinuity it allows us to find curves of shortest lengthbetween a pair of points contained in a sufficiently large compact sub-set. Here and later we denote by Br(x) the open metric ball of radiusr around x:

Proposition 2.1. Let the closure Br(x) be compact. Let y ∈ Br(x) bea point connected to x by a curve of length ≤ r. Then there exists acurve of shortest length among all curves connecting x and y in X.

2.3. Length spaces. The metric space (X, d) is called a length spaceif, for all x, y ∈ X

(2.1) d(x, y) = inf`(γ) | γ : [a, b]→ X, γ(a) = x, γ(b) = y

Note that inequality ” ≤ ” is true for all curves γ connecting x and y.We also call d a length metric or intrinsic metric.A few (hopefully well-known) classical examples:1) The Euclidean space, and its convex subsets are length spaces.2) R2 \ 0 is a length space but not R2 without a segment. Here we

consider the subset with the metric restricted from the ambient spaces.3) Any Riemannian manifold (always connected, by default) (M, g),

in particular, a surface M2 in R3 with the first fundamental form g, isa length space with respect to the Riemannian metric dg induced by g.

4) Any normed vector space with the metric induced by the normand any convex subset of a normed space is a length space.

2.4. First properties. A shortest curve γ : [a, b] → X between γ(a)and γ(b) in a length space X satisfies d(γ(t), γ(s)) = `(γ|[s,t]), for alls < t ∈ [a, b]. If, in addition, γ is parametrized by arclength, thend(γ(s), γ(t)) = |t− s|, for all s < t ∈ [a, b].

Shortest curves play in metric geometry the role of segments in Eu-clidean geometry. They will be parametrized by arclength, if not oth-erwise stated. Often, shortest curves in metric geometry are calledgeodesics by an abuse of notation.

Recall that shortest curves in Riemannian geometry solve the geo-desic equation, but solutions of this equation are shortest curves onlyon sufficiently short subintervals.

If X is a locally compact length space then the theorem of Arzela-Ascoli stated above provides the following statement. For every pointx ∈ X there exists some r > 0 such that for every pair of pointsy, z ∈ Br(x) there exists a shortest curve in X connecting y and z.

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Note, that such a shortest curve is contained in B3r(x). In general,such a curve is not contained in Br(x), thus the ball Br(x) is in nosense convex.

In general, there is no kind of uniqueness for shortest curves withprescribed fixed points, even locally.

In length spaces the closure Br(x) of the open ball is the closed ball,thus the set of points y ∈ X such that d(x, y) ≤ r.

A pointwise limit of shortest curves in a length space is a shortestcurve and its limit is the limit of lengths of the approximating curves.

An important consequence of the theorem of Arzela-Ascoli is thefollowing theorem of Hopf–Rinow and Cohn-Vossen, which might beknown from a course in differential geometry.

Theorem 2.2. Let X be a locally compact length space. Then X isa complete metric space if and only if all of its bounded closed subsetsare compact. In this case any pair of points in X is connected by ashortest curve.

3. Constructions of length spaces

3.1. Induced length metric. Let (X, d) be a metric space, let A ⊂ Xbe a subset. We say that A is rectifiably connected if for any x, y ∈ Athere exists a curve of finite length γ : I → A connecting x and y.

The induced intrinsic metric dA on a rectifiably connected subsetA ⊂ X is defined as

dA(x, y) = inf`(γ) | γ : [a, b]→ A, γ(a) = x, γ(b) = y .On A we have the inequality dA ≥ d.

For A = X the equality dA = d is equivalent to the statement thatd is a length metric.

The following Lemma is elementary but non-trivial.

Lemma 3.1. For any rectifiably connected subset A ⊂ X, the metricdA is a length metric on A. Moreover, for any curve γ : I → A, thelength of γ with respect to d and dA coincide.

The main classical example of this construction is the induced in-trinsic metric on a connected submanifold of Rn. This is nothing elseas the Riemannian metric induced by the first fundamental form. Theinvestigation of such metric spaces is the central subject of Riemanniangeometry.

3.2. Cartesian product. Given metric spaces (X, dX) and (Y, dY ) wedefine the product metric on the Cartesian product

Z = X × Y = (x, y) | x ∈ X, y ∈ Y 6

by the formula of Pythagoras

d((x1, y1), (x2, y2)) =√d2X(x1, x2) + d2

Y (y1, y2) .

We say that a curve γ : I → X in a metric space X has constantvelocity v ≥ 0 if, for all s, t ∈ I,

d(γ(s), γ(t)) = v · |t− s| .Any curve of finite length can be reparametrized to have constant ve-locity v.

We observe that if γ1 : I → X is a curve of constant velocity v1 andγ2 : I → Y is a curve of constant velocity v2 then γ = (γ1, γ2) : I →X × Y is a curve of constant velocity

√v2

1 + v22.

Now we can easily see:

Lemma 3.2. If X and Y are length metric spaces then X × Y is alength metric space as well. An arclength parametrized curve in X×Yis a curve of shortest length if and only if its projections to X and Yare curves of shortest length parametrized proportionally to arclength.

Proof. Given two points z1 = (x1, y1) and z2 = (x2, y2), we find curvesγ1 in X and γ2 in Y connecting x1 to x2, respectively, y1 to y2, suchthat `(γ1) is arbitrary close to d(x1, x2) and `(γ2) is arbitrary close tod(y1, y2).

We now reparametrize γi to be defined on [0, 1] and to have constantvelocity `(γ1). Then γ = (γ1, γ2) connects z1 and z2 and has constant

velocity√`2(γ1) + `2(γ2).

Choosing γi to (almost) realize the distance between x1 and x2, rep-sectively, between y1 and y2, we see that `(γ) gets arbitrary close tod(z1, z2). This shows the first statement.

The remaining statements are similar and left to the reader.

3.3. Euclidean cone. Let X be a metric space. The Euclidean coneCon(X) over X is defined as follows. As set Con(X) is

(X × (0,∞)) ∪ o = (X × [0,∞))/X × 0 .Elements of Con(X) are written as (x, t) =: t ·x, with t ≥ 0 and x ∈ X.All points 0 · x are identified to one point o, the origin of the cone.

We define the metric on Con(X) in two steps. In the first step wetruncate the metric on X by declaring

d(x, y) = mind(x, y), π .Thus d and d coincide locally around any point.

We think of X as sitting in the unit sphere, of the distance in X asthe ”angle” and the first step just cuts all the angles by π.

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Now we define the cone metric dc on Con(X) as follows. For anypoints (x, t) and (y, s) in Con(X) we find a pair of points x and y in

S1 ⊂ R2 such that d(x, y) = dS1(x, y).

Then we consider the points t · x and s · y in R2 and set

dc((x, t), (y, s)) := |t · x− s · y| .In particular, dc(o, (x, t)) = t for all (x, t) ∈ Con(X).We have a canonical embedding of X via x → (x, 1) onto the unit

distance sphere in Con(X) around the origin. Under this identification

dc(x, y) corresponds to d(x, y) as arclength to chord-length on the circle.More precisely,

dc(x, y) = 2 · sin(d(x, y)

2) .

By constructionCon(Sn−1) = Rn ,

if Sn−1 is considered with its natural (= of constant curvature one =induced intrinsic) metric.

Any map f : X → Y extends to a natural map Con(f) : Con(X)→Con(Y ) by

Con(f)(tx) := t · f(x)

If f preserves the distances then so does Con(f) and if f is 1-Lipschitzthen so is Con(f). The last statement is an elementary (but not abso-lutely trivial) statement in Euclidean geometry.

Embedding an interval isometrically into S1 we see

Lemma 3.3. If X is isometric to an interval of length α ≤ π thenCon(X) is isometric to the convex subset in R2 bounded by two raysenclosing angle α.

Moreover, by definition, or looking on S0, we see that for any pointsx, y ∈ X with d(x, y) ≥ π the cone Con(x, y) = R is isometricallyembedded into Con(X).

Now we can show:

Proposition 3.4. Con(X) is a metric space, which is a length spaceif so is X.

Proof. Clearly dc is symmetric, non-negative and vanishes only betweenidentical points.

In order to prove the triangle inequality, consider zi = (ti · xi) ∈Con(X) for i = 1, 2, 3. We then find a map f : x1, x2, x3 → S1 sothat for the images xi = f(xi) we have

d(x1, x3) = dS1

(x1, x3) ; d(x1, x2) ≥ dS1

(x1, x2) ; d(x2, x3) = dS1

(x2, x3) .8

Looking at Con(f) and the triangle inequality in R2 we see

dc(z1, z3) = dc(z1, z3) ≤ dc(z1, z2) + dc(z2, z3) ≤ dc(z1, z2) + dc(z2, z3) .

This shows that Con(X) is a metric space.Assume now that X is a length space. Let zi = (ti, xi) ∈ Con(X) be

as above. If t1 · t2 = 0 or d(x1, x2) ≥ π then z1 and z2 are connected bya radial shortest curve with length d(z1, z2) = t1 + t2.

If d(x1, x2) < π, consider a shortest curve γ : I → X if it existsconnecting x1 and x2. If it does not exist apply the subsequent argu-ment to almost shortest curves between x1 and x2. We may assumethat γ is parametrized by arclength, thus γ : I → X is an isometricembedding and so is Con(γ). But Con(I) is a convex subset of R2,which is a geodesic spaces. Connect preimages of zi in Con(I) by a(unique) shortest curve in Con(I) and sent it by Con(γ) to Con(X) toobtain a shortest curve between z1 and z2 of the right length.

3.4. Quotient spaces. Let X be a metric space and let ∼ be an equiv-alence relation on X.

Define the function d∼ : X ×X → [0,∞) as follows.

d∼(x, y) = inf∑

d(qi, pi+1) ,

where the infimum is considered over finite sequences p1, q1, p2, q2, ..., pm, qmsuch that x = p1, y = qm and pi ∼ qi for all i.

Then d∼ ≤ d, d is symmetric and satisfies the triangle inequality.Moreover, d∼(x, y) = 0 if x ∼ y.

(It is easy to see, that d∼ is the largest function with these threeproperties).

We set x ≡ y if d∼(x, y) = 0. It is again an equivalence relation onX, coarser than the equivalence relation ∼.

Define the space X/ ∼ as the set of equivalence classes [x] of the

relation ≡ with the metric d∼([x], [y]) := d∼(x, y). We have:

Lemma 3.5. The space X/ ∼ is a metric space. The canonical mapx→ [x] from X to X/ ∼ is 1-Lipschitz. If X is a length space then sois X/ ∼.

Proof. All statements but the last one follow directly by construction.To find a curve of short d∼-length between [x] and [y], we consider asequence pi, qi as in the definition of d∼(x, y) and connect qi with pi+1 bya curve γi of short length in X. Now observe, that the concatenationof the projections of the curves γi to X/ ∼ is a continuous curve inX/ ∼, whose length can be chosen arbitrary close to d∼(x, y).

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The definition seems complicated but the construction should beintuitively clear, at least in most important examples. Here are somecomments and examples.

1) Let A ⊂ X be a subset. Consider ∼:=∼A by letting two pointsbe equivalent iff they are equal or both contained in A. Then

d∼(x, y) = mind(x, y), d(A, x) + d(A, y) .A very simple (but already very strange) example arises if X = R2 andA = B1(0). Then X/ ∼A is a space homeomorphic to R2 and locallyisometric to R2 outside one ”thick point”, the image of A. Note thatany curve which runs around this point (remember from topology orcomplex analysis what it means) has length at least 2π!

2) It can happen that d∼(x, y) = 0 also if x and y are not equivalentwith respect to ∼. In other words, ∼6=≡. For example, it happens ifA in the previous example is not closed in X.

3) Let X = [0, 2π] and let A = 0, 2π in the first example. ThenX/ ∼A= S1.

4) For X = [0, 1] × [0, 1] and the equivalence relation ∼ identifyingopposite sides of X, it can be shown (Exercise!) that X/ ∼ is the directproduct S1 × S1

5) Let X1, X2 be metric spaces, let A be a subset of X1 and letφ : A → X2 be a map. Consider the metric space X given as thedisjoint union of X1 and X2, where we define the distance betweenpoints in the same Xi as in original space Xi and between x1 ∈ X1 andx2 ∈ X2 to be infinity (by an abuse of definition). Consider now on Xthe equivalence relation ∼φ, where the only non-singleton equivalenceclasses are subsets f−1(x2) ∪ x2 for x2 ∈ φ(A1).

Let us explore the space X/ ∼φ in some special cases:If X2 is a point, then X/ ∼φ is the space X/ ∼A from example (1).If A = X1 and φ is 1-Lipschitz then X/ ∼φ is isometric to X2.Let now φ : A→ φ(A) ⊂ X2 be isometric. Often, one would call the

arising space X/φ the gluing of X1 and X2 along the isometry φ. Inthis case, the natural projection from X to X/ ∼φ sends X1 and X2

isometrically onto their images.For x1 ∈ X1 and x2 ∈ X2 the distance satisfies

d∼(x1, x2) = infa∈A

(d(x1, a) + d(x2, φ(a)) .

4. Examples of gluings. Simplicial complexes

4.1. Observation about quotient spaces. While it is often intu-itively clear, what a quotient space, repsectively a gluing of spacesshould look like, the formal verification is often tedious. Here are a few

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helpful observations. We restrict ourselves mostly to length spaces,even to the most important case of such.

We call a length space X a geodesic space if any pair of points ofX are connected by a shortest curve. Recall that by the theorem ofCohn-Vossen and Hopf–Rinow, every complete, locally compact lengthspace is geodesic.

Lemma 4.1. Let X, Y be geodesic metric spaces, let f : X → Y bea surjective map. Denote by ∼=∼f the equivalence relation on X,whose equivalence classes are exactly the fibers of f . Then f inducesan isometry f : X/ ∼f→ Y if the following conditions hold true.

(1) f is 1-Lipschitz(2) For every shortest curve γ : [a, b]→ Y there exist a = t1 ≤ t2 ≤

... ≤ tm = b and lifts of restrictions γ|[ti,ti+1] to shortest curvesηi : [ti, ti+1] → X of the same length. Thus, `(ηi) = `(γ|[ti,ti+1])and f ηi = γ|[ti,ti+1].

Proof. Since f is 1-Lipschitz, the definition of the distance d∼ in X andthe triangle inequality in Y imply that

d∼(x, x′) ≥ dY (f(x), f(x′)) ,

for all x, x′ ∈ X. Therefore, f(x) = f(x′) if and only if d∼(x, x′) = 0.Thus, f induces a bijective 1-Lipschitz map f : X/ ∼f→ Y .

Now consider arbitrary y, y′ ∈ Y connect them by a geodesic γ.Applying the second condition, subdivide γ and take pi = ηi(ti) andqi = ηi(ti+1). Then, by definition, we see that x = f−1(y) and x′ =f−1(y′) satisfies

d∼(x, x′) ≤ `(γ) = d(y, y′) .

Thus, f is an isometry.

For example, this lemma allows us to verify directly the intuitivelyclear Example (3) in the previous section. It also shows the following.If X is a convex sector in R2 of angle α and ∼ identifies all pairs ofpoints on the boundary rays of X at the same distance from the origin,then the arising space is the Euclidean cone over a circle of length α.

Another useful observation says that under some often encounteredassumptions, one obtains a small neighborhood in the quotient spaceby the quotient of corresponding small neighborhoods. Here is a moreprecise statement, the proof is much less complicated than the state-ment and is left to the reader.

Lemma 4.2. Let X be a metric space and ∼ be an equivalence relation.Let x ∈ X and ε > 0 be given. Assume that the ε-neighborhood Bε in

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X of the equivalence class [x] of x is a union of equivalence classes.Assume that for all z, z′ ∈ Bε with z′ ∼ z we have d([x], z) = d([x], z′).

Then the ball B ε2([x]) in X/ ∼ is isometric to the quotient B ε

2/ ∼ of

the ε2-neighborhood B ε

2of [x] in X.

4.2. Metric graphs. Let Ii be a disjoint union of compact intervals(for simplicity, finite or denumerable). Let ∼ be some equivalencerelation on the union of boundary points of the intervals. We (usually)require, that the endpoints of the same interval are not in the sameequivalence class.

Define now a space by gluing the intervals along the equivalencerelation. Thus, we consider first the disjoint union of the intervals withthe distance between points in different intervals set to be infinity.Then use the equivalence relation and obtain a quotient space. Thearising metric space is called a metric graph. We will mostly restrictourselves to one connected component of it.

In a metric graph, we have edges and vertices. Any point in theinterior of an edge has a neighborhood isometric to an open interval.We have a canonical embedding of any Ij into the arising metric graphZ. Note that the embedding is length preserving but it may not be anisometry (example?!).

Assume now that all intervals Ii which are adjacent to a vertex xhave lengths bounded from below. Then a neighborhood of this pointx in Z is isometric to a small ball in a gluing Y of a number of raysalong the common endpoint. In other words, Y is the cone Con(X)over a discrete metric space X at which all points have distance π fromeach over.

On the other hand, it is not very difficult to prove:

Proposition 4.3. Let X be a locally compact length metric space inwhich every point has a neighborhood isometric to a ball in the coneover a finite space. Then X is a metric graph.

4.3. Simplical complexes in higher dimensions. A general metricsimplicial complex is defined similarly to metric graphs.

We consider a disjoint union of convex Euclidean simplices ∆i (pos-sibly of different dimensions). For any two different simplices i, j werequire the existence of some (possibly empty) face Ai of ∆i and Aj of∆j and an isometry φij : Ai → Aj. Note, that a face here mean a faceof any codimension. It can be the empty set, a vertex, an edge, .... ,or the whole simplex.

Now we construct a space X by gluing the simplices ∆i along theisometries φij.

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We can get a good feeling of the construction by gluing the bound-ary of tetrahedron from 4 triangles or the boundary of cube from 12triangles.

Very often (always for the rest of this course) we will require thatat any point only finitely many triangles are glued together, in otherwords, any equivalence class arising in the course of the constructionis finite.

For any point in any simplex, any sufficiently small ball around anypoint is isometric to a ball in a Euclidean cone. This (and some notquite trivial thoughts) show that in any metric simplical complex asmall ball around any point is isometric to a ball in some Euclideancone. In fact there is a (highly non-trivial) converse:

Theorem 4.4. Let X be a locally compact, complete length space. As-sume that any point in X has a neighborhood isometric to a ball insome Euclidean cone. Then X is isometric to a simplicial complex.

In the lecture, I will explain the proof of the theorem only in thefollowing special situation. A simpicial surface is a locally compactmetric space, in which every point has a neighborhood isometric to aball in a cone over some circle or in a cone over an interval.

With a little knowledge in algebraic topology, it is possible to prove,that a metric simplicial complex is a simplicial surface if and only ifit is homeomorphic to a two-dimensional manifold (with boundary),thus if every point has a neighborhood homeomorphic to plane or to ahalfplane.

4.4. Total angles. Let X be a simplical surface. For any point x ∈ Xa small ball Bε(x) is isometric to the ε-ball in Con(Σx), where Σx is acircle or an interval. The isometry type of Σx is uniquely determined,for instance by the topology and lengths of small distance circles aroundx in X. We will call Σx the space of directions in x. Its length will becalled the total angle at x.

The set of points where Σx is an interval is a closed subset, theboundary ∂X of X. The complement is the interior of the surface

For all but discretely many points x ∈ ∂X, the total angle at x is πand for all but discretely many points in X \ ∂X the total angle at xis 2π.

The corresponding points will be called the (interior, respectivelyboundary) vertices of the surface. At each interior vertex the valueω(x) = 2π− `(Σx) will be called the curvature at x. At each boundaryvertex the value κ(x) = π−`(Σx) will be called the turn of the boundaryat x.

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For instance, consider the boundary X of the tetrahedron. It is asurface without boundary. At each vertex of the tetrahedron the totalangle is π, hence the curvature is π as well.

Consider now the surface Y of the tetrahedron with one face deleted.Then there remains one interior vertex with total angle π and 3 bound-ary vertices with turn π

3at each of them.

5. Gauß–Bonnet in simplicial surfaces (05.05)

5.1. Signed measures. Here and later a signed measure ω on a locallycompact topological space X is a difference of two Borel measures ω1−ω2 each of them being finite on compact subsets. A Borel functionf : X → R is integrable with respect to ω if it is integrable withrespect to ω1 and ω2. And we set∫

f dω :=

∫f dω1 −

∫f dω2 .

In particular, for any Borel subset A contained in a compact subset Kof X the value

ω(A) = ω+(A)− ω−(A) =

∫X

χA dω

is well defined.All continuous functions with compact support are integrable. It is

a deep theorem of Riesz, maybe discussed in a lecture on functionalanalysis, that the set of signed measure is exactly the dual space (withrespect to the right topology) of the topological vector space of contin-uous functions with compact support.

Another non-trivial related statement is that any signed measure ωcan be uniquely represented as a difference of two measures ω = ω+−ω−with the following property. There exist a Borel decomposition X =A+ ∪ A− such that ω+(A−) = ω−(A+) = 0.

These measures ω± are called the positive and the negative part ofthe signed measure ω. The measure

|ω| := ω+ + ω−

is called the variation of the signed measure ω.The principal example one should have in mind is the following. Let

µ be some Borel measure on X, which is finite on compact subsets. Leth : X → R be a Borel function, which is integrable over any compactsubset. Then we have the signed measure h · µ on X such that for any

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Borel subset A contained in a compact subset of X

h · µ(A) =

∫X

χA d(h · µ) =

∫A

(h) dµ .

Setting A+ = h−1([0,∞)) and A− := h−1((−∞, 0)) we have a disjointdecomposition of X as X = A+ ∪ A−. The positive and the negativeparts of h · µ are exactly the measures

(h · µ)±(B) :=

∫B∩A±

h .

The variation of h · µ is the measure |h| · µ.Indeed, every signed measure arises (non-uniquely) in the way de-

scribed above. Indeed, any signed measure ω can be written as h · |ω|,where h = χA+ − χA− .

5.2. Curvature and turn measures of a simplicial surface. LetM be a simplicial surface with (possibly empty) boundary ∂M . We de-fine two signed measures on M , the curvature measure ω, concentratedin M \ ∂M and the turn κ concentrated on ∂M .

For any subset B contained in a compact subset of X we set

ω(B) :=∑x∈B

ω(x) ,

where ω(x) is the curvature at the point x ∈ X, thus the differencebetween 2π and the total angle in x. We obtain the positive and thenegative part ω± by restricting in the above formula the summation toall points in B with positive, respectively, negative curvature.

Note that the set V of points in X \ ∂X with ω(x) 6= 0 is discrete inX. Thus the following sum of Dirac measure

µ :=∑x∈V

δx

is a locally finite (counting) measure on X. The curvature measure ωis nothing but ω · µ, where we see ω as a locally integrable functionwith respect to µ.

We call ω(B) the curvature of the set B and |ω|(B) the absolutecurvature of B.

Similarly, we define the turn as a measure concentrated on all pointsv ∈ ∂X with non-zero turn κ(v). Again, for any subset B contained ina compact subset of X we define the turn of the boundary of B as

κ(B) :=∑

v∈B∩∂X

κ(v) .

By construction |κ|(X \ ∂X) = |ω|(∂X) = 0.15

5.3. Gauß–Bonnet in simplicial surfaces. We can now formulatethe global theorem of Gauß– Bonnet for simplical surfaces.

Theorem 5.1. Let M be a compact simplical surface with possiblyempty boundary ∂M . Let ω denote the curvature measure and let thesigned measure κ denote the turn of the boundary. Then

ω(M) + κ(∂M) = 2πχ(M) .

The proof was explained in the first section for the case of emptyboundary. The same proof applies here. We write M as any simplicialsurface, as a gluing of Euclidean triangles. The sum s of all angles of alltriangles is exactly π ·F , where F is the number of triangles. Summingthe angles ”‘vertice-wise”, we get s = sint + sbound, with

sint =∑

v∈M\∂M

(2π − ω(v)) and sbound =∑v∈∂M

(π − κ(v)) .

We arrive at

π · F = 2π · Vint − ω(M) + π · Vbound − κ(∂M) ,

where Vint and Vbound denote the number of interior, respectively bound-ary vertices of M . This shows

ω(M) + κ(∂M) = 2π · V − π · Vbound − π · F .

It is now a combinatorial exercise to see that π · Vbound + π · F equals2πE − 2πF , with E the number of edges. This finishes the proof.

5.4. Local version of Gauß–Bonnet. Let again M be a simplicialsurface, let x ∈ M be a point and let Γ be the concatenation of twoshortest curves starting in x. A small ball B around x is subdividedby Γ into two parts Ul and Ur (the ”left” and ”right” connected com-ponents). The closure of this two parts in B is exactly Ul,r ∪ (Γ ∩ B).These are two simplicial surfaces with boundary Γ ∩ B. There are nointerior vertices in Ul,r and at most one boundary vertex x. We havethe turns κl(x) of Γ at x with respect to the surrounding manifold Ul,r.

By construction the sum of the left and the right turns κl(x) +κr(x)is exactly the curvature ω(x) in the original space.

Let now U ⊂ M be an open subset homeomorphic to R2 and letΓ ⊂ U be a simple closed curve. Assume that Γ is a polygon, thus aconcatenation of shortest curves.

By the Jordan curve theorem (see next section), there exists a uniqueconnected component O of U \ Γ homeomorphic to a an open disc.Moreover, O = O ∪ Γ is homeomorphic to a closed disc.

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The set O in its intrinsic metric is a simplical surface with boundaryΓ. At any point x ∈ Γ we have the turn κ(x) of Γ with respect to Oand the theorem of Gauß–Bonnet applied to O gives us

ω(O) + κl(Γ) = 2π ,

where κl(Γ) is the turn of the polygon Γ inside O.An important point of caution: Unlike in Riemannian geometry,

even for a triangle Γ, the contribution of the sides of the triangles tothe turn of Γ may be non-trivial. The reason is that a shortest curvein M can pass through vertices v of M . However, this can only happenif ω(v) ≤ 0. Also the contribution of such a point to the turn cannotbe positive.

6. Topology of surfaces (07.05)

6.1. Jordan, Schoenflies. In this and the next subsection we discusssome classical results in the theory of surfaces. The proofs are too longand too complictaed to be included here.

The following seemingly trivial theorem, a strengthening of the the-orem of Jordan, due to Schoenflies, is surprisingly difficult to prove.

Theorem 6.1. Let Γ be a simple closed curve in R2. Then there existsa homeomorphism F : R2 → R2 which sends Γ onto S1.

In particular, R2 \ Γ has two components, one bounded and one un-bounded. Both components have Γ as their boundary. The closure ofthe bounded component is homeomorphic to a closed disc.

Also in contrast to higher dimensions simple non-closed curves can-not be wild. This can be deduced by a trick from the above theorem:

Korollar 6.2. For any compact simple curve Γ in R2 there exists asimple closed curve Γ containing Γ. There exists a homeomorphismfrom R2 to R2 which sends Γ to the interval [0, 1] ∈ R.

Thus, whatever we can describe in topological terms for the circleor a segment in R2 we can transform to any arc and an simple closedcurve, called a Jordan curve.

For instance, any arc locally ”‘separates”’ a neighborhood in twoparts and we can talk about convergence of points or curves from agiven side to the arc. In other words, we can locally distinguish leftand right sides of the arc (without knowing which side is left and whichis right). In order to fix right and left we would need an orientation ofR2 and of the arc.

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6.2. Topological surfaces. A surface with boundary is a metric (orjust metrizable) topological space in which any point has a neighbor-hood homeomorphic to a plane or to a halfplane. Based on the theoremof Schoneflies the following result has been proved by Rado:

Theorem 6.3. Any surface admits a triangulation. In other words,any surface is homeomorphic to some simplicial surface.

The triangulation is definitely not unique, but for compact surfaces,the Euler characteristic of any simplicial surface homeomorphic to itis a topological invariant. Thus χ(M) is well-defined. It essentiallydetermines all of the topology of M as follows from the classificationof surfaces:

Theorem 6.4. Two compact surfaces M1,M2 are homeomorphic if andonly if they have the same number of boundary components, the sameEuler characteristic and are both orientable or non-orientable.

From the theorem about the existence of the triangulation or fromthe classification result above the following consequence can be derived:

Theorem 6.5. Any topological surface admits a smooth atlas. It ad-mits a smooth Riemannian metric and a geodesic triangulation, suchthat the boundary is a disjoint union of geodesic polygons.

7. Comments on convex simplicial surfaces (07.05).

The following result might be intuitively clear. We explain the ideaof the proof, skipping some details

Proposition 7.1. Let X 6= R3 be a convex cone in R3, thus a closedconvex subset invariant under multiplication with non-negative num-bers. Let Y be its boundary equipped with the induced intrinsic metric.Then Y is a Euclidean cone over a circle of length ≤ 2π.

Proof. Y with the restricted metric is a Euclidean cone Con(Z), whereZ is the intersection of Y with the unit sphere. Hence, Y is a Euclideancone with respect to the induced intrinsic metric and we only need toshow that Z has length ≤ 2π (Exercise).Z is the boundary in S2 of the convex subset X ⊂ S2. Approxomat-

ing Z by a polygon, it suffices to prove that a polygonal boundary ofa convex subset of S2 has length at most 2π. This is shown by fixingan upper bound on the number of sides, maximizing the length andproving that this maximal object must have length 2π.

A direct consequence of this statement is:18

Korollar 7.2. Let M be the boundary of a convex polyhedron in R3.Then M is a simplical surface homeomorphic to S2 and the curvaturemeasure ω on M is non-negative.

This is the discrete analogue of the result well-known from a lecturein differential geometry saying that a smooth convex surface in R3 hasnon-negative Gauß curvature everywhere.

The converse is a deep theorem of Alexandrov. Namely, every non-negatively curved compact simplical surface homeomorphic to S2 isisometric to the boundary of a convex polyhedron in R3 uniquely de-termined up to rigid motions.

This result together with the observations that any convex body inR3 can be approximated by convex polyhedra establishes a very preciseconnection between smooth surfaces of non-negative curvature, convexbodies in R3 and simplicial surfaces of non-negative curvature. Thisconnection lies in the origin of the theory of surfaces with boundedintegral curvature, which we want to approach soon.

8. Angles in metric spaces (12.05)

8.1. Definition of angles. For any 3 points A,B,C in a metric spaceX there exists a triangle A, B, C ∈ R2 with side lengths equal tod(A,B), d(B,C), d(C,A), which is unique up to motions of the Eu-clidean plane. It is called the comparison triangle of the triangle (herejust 3 points) A,B,C.

The comparison angle ∠A(B,C) ∈ [0, π] is the angle at A of thiscomparison triangle.

Let now X be a length space and let γi : [0, εi] → X for i = 1, 2 beshortest curves starting in the same point x = γi(0). For any s, t > 0,

consider the comparison angle α(s, t) := ∠x(γ1(s), γ2(t)) and define theupper angle ∠(γ1, γ2) between γ1 and γ2 as

∠(γ1, γ2) := lim sups,t→0

α(s, t) .

We say that the angle between γ1 and γ2 is well-defined (and equals∠(γ1, γ2)) if the lim sup in the definition is really a limit.

First, a simple example. In the Euclidean cone Con(Σ) over any(length) space Σ the angle between any shortest curves γ1, γ2 startingat the origin o of Con(Σ) are well-defined. The curves γi have theform γi(t) = t · xi for some xi ∈ Σ and the angle between γ1 and γ2

equals minπ, dΣ(x1, x2), by the construction of the Euclidena cone.In particular, this applies to all simplical metric spaces at all points ofthe spaces.

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The following example is much less trivial and requires some knowl-edge in Riemannian geometry. If M is a Riemannian manifold andγ1 and γ2 are arbitrary shortest curves starting in a point x, then γihave well-defined directions vi ∈ TxM and the angle between γ1 andγ2 is exactly the angle between the vectors v1 and v2 in the Eucldieanspace TxM . (Exercise: try to prove it!). Note that the local theoremof Gauß–Bonnet discussed in the first lecture was using exactly thisnotion of angles.

The last statement becomes much easier (Exercise: try to provethis!), if in the definition of angle one does not consider arbitrary limitss, t→ 0 but only such limits for which s/t remains bounded away from0 and∞. This would provide an alternative definition of (upper) anglesand everything what follows would work with that definition as well.

The next example is even less trivial and requires some knowledge inconvex geometry. Let X be a normed vector space. The angle betweenany pairs of shortest curves starting in the same point exist if and onlyif the norm comes from a scalar product (Exercise: try to prove this,at least check it for some examples).

8.2. Space of directions. Let X be a length space, x ∈ X a point.Consider the set Γx of all shortest curves starting in x. Then the upperangle ∠ : Γx × Γx → [0, π] is symmetric, vanishes on the diagonal andsatisfies the triangle inequality (Exercise: Prove that!).

Identify all shortest curves which include an upper angle of 0 anddenote by Σx = Γx/ ∼ the space of all such equivalence classes. Notethat ∠ defines now a metric on Σx. Either the space Σx or its metriccompletion Σx is called the space of directions of the space X at thepoint x

From the above examples, we see that for a Euclidean cone Con(Σ)the space of directions Σo at the origin o of the cone is identified withthe space Σ on which the metric is truncated by π.

For a Riemannian manifold M and any point x ∈ M the space ofdirections Σx = Σx is canonically identified (what does this mean?)with the unit sphere Sn−1 in the Euclidean space TxM .

An easy example of a compact space for which the space of directionsΣx at some point x is not complete is the closed unit disc D ⊂ R2

(Question: why?). Answering this question, you might also recognize inthis example, why it might be more natural to consider the completionΣx instead of Σx.

8.3. First formula of variation. Recall that for a point x ∈ R2,and a shortest curve γ(t) = y + t · v starting at a point y 6= x the

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function f(t) = d(x, γ(t)) satisfies f ′(0) = − cos(α), where α is theangle between γ and the segment [yx] at the point y (Exercise: Provethat). Remember, that a similar statement holds in any Riemannianmanifold (Question: what statement?).

It turns out that half of this formula holds true in any length metricspace:

Proposition 8.1. Let x, y be points in a length space X. Let γ be anyshortest curve connecting y and x. Let η : [0, ε] → X be any shortestcurve starting in y. Set f(t) = d(x, η(t)). Then the 1-Lipschitz functionf : [0, ε]→ R satisfies

f+(0) := lim supr→0

f(r)− f(0)

r≤ − cos(∠(γ, η)) .

Here is a sketch of the proof (Exercise: try to fill in the details!): Forall small s we have

f(r)−f(0) ≤ (d(x, γ(s))+d(γ(s), η(r))−(d(x, γ(s))+s) = d(γ(s), η(r))−s .Now, for all s, r, sufficiently small, the comparison angle ∠x(γ(s), η(r)))is almost bounded from above by ∠(γ, η). Now, if r is much smaller thans we can estimate s− d(γ(s), η(r)) by the first formula of variation inthe Euclidean comparison triangle.

9. Main definition (14.05)

9.1. Excess of a triangle. Let X be a length space. A triangle in Xconsists of 3 points A,B,C and three shortest curves [AB], [BC], [CA]connecting them. Note that the sides [AB], [BC], [AC] are not uniquelydetermined by the vertices A,B,C. Nevertheless, we will denote thetriangle consisting of these three sides by ABC.

Let X be locally compact, and U be an open subset of X, such thatU is compact. For all points A,B,C ∈ U whose pairwise distancesare smaller than the distance d(A,B,C, ∂U), the points A,B,C de-termine (at least) one triangle ABC. Moreover, any such triangle iscontained in U (Exercise: Prove the last statements).

Denote by α, β, γ the upper angles of the triangle ABC at A,B,C.Thus, α is the upper angle between the sides [AB] and [AC] at A andso on. The upper excess of the triangle ABC is the quantity

δ(ABC) := α + β + γ − π .Let α0, β0, γ0 be the comparison angles: α0 = ∠A(B,C) and so on.Then (why?)

δ(ABC) = (α− α0) + (β − β0) + (γ − γ0) .21

9.2. Some notions of convexity. A subset G of a length space Xwill be called convex, if any two points in G are connected in G by ashortest curve. The subset G is called completely convex if, in addition,all shortest curves in X between any pair of points in G is contained inG. (Exercise: what are examples of convex but not completely convexsubsets? Can the whole space X be non-convex?)

(Exercise: An intersection of completely convex subsets is completelyconvex. An intersection of convex subsets may fail to be convex).

Let X be a length space homeomorphic to a surface, let U ⊂ X bean open subset homeomorphic to a disc. Let Γ be a simple closed curveof finite length in U . Let finally C be the closed disc in U bounded byΓ, the Jordan domain of C. We say that C is convex relative to theboundary if the following conditions hold true.

(1) The distance from C to ∂U is at least 4 times the length of Γ,also denoted as the perimeter of C.

(2) For any pair of points A,B on Γ, any simple arc Γ′ outside ofC connecting A and B in U divides Γ in two arcs Γ+ and Γ−. One ofthese parts, say Γ+ is closer to Γ (what does it mean precisely?). Wenow require, that for any such arc Γ′, the length of Γ′ is not smallerthan the length of arc Γ+.

Any subset convex relative to its boundary is convex (Exercise!).But it may not be completely convex (Example?). It is slightly morecomplicated to find examples showing that a completely convex subsetC bounded by a rectifiable curve may violate the second conditionabove.

We end this subsection with two examples. Assume first that U isa completely convex open disc in X. Assume moreover, that shortestcurves between any pair of points in U are uniquely determined by theendpoints. (Recall that such an U exists if X has a Riemannian metric.Prove, using Gauß–Bonnet that the uniqueness statement is true if Xis a polyhedral surface and the curvature measure is non-positive).

Then a subset C of U is convex if and only if it is completely convex.Moreover, if a Jordan curve Γ in U bounds a convex subset C andd(C, ∂U) ≥ 4·`(Γ) then C is convex relative to the boundary (Exercise,several steps are needed for the solution).

In the second example we consider the cone X = Con(Σ) over acircle of length 2α < 2π. Take a triangle ABC, with A the vertex ofthe cone d(A,B) = d(A,C) and the (comparison=upper) angle at Aequal to α. Then the triangle is convex but not completely convex norconvex with respect to the boundary.

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9.3. Simple triangles in surfaces. Let a length space X be home-omorphic to a surface. We call a triangle T = ABC a simple triangleif the sides of T form a simple closed curve which is contained in anopen subset U homeomorphic to a disc. Moreover, we require T to beconvex relative to its boundary in U .

For example, any sufficiently small non-degenerated in any Riemann-ian surface is simple, if it is not contained in a single geodesic.

In a cone over a circle of length ≥ 2π any triangle is simple if the sidesintersect only in vertices (Exercise: why is the additional conditionneeded?). In a cone over a circle of length < 2π there are many non-simple triangles

9.4. Main definition. We call two simple triangles in a length spaceX homeomorphic to a surface non-overlapping if their interiors (theirJordan domains) do not intersect.

We say that a length space X homeomorphic to a surface is a surfaceof bounded curvature if any point x ∈ X admits an open neighborhoodU homeomorphic to a disc and a constant Ω(U) > 0 with the followingproperty. For any finite sequence of non-overlapping simple trianglesT1, ..., Tm ⊂ U , we have

δ(T1) + ....+ δ(Tm) ≤ Ω(U) .

Note, that we only assume a one-sided bound on the sum of the defects.Rather surprisingly the opposite bound turns out (much much later)to follow as a consequence.

From the theorem of Gauß–Bonnet we deduce that any Riemanniansurface and any simplicial surface have bounded curvature in the senseabove (Exercise).

References

[BBI01] D. Burago, Y. Burago, and S. Ivanov. A course in metric geometry, vol-ume 33 of Graduate Studies in Mathematics. American Mathematical So-ciety, Providence, RI, 2001.

[Res93] Yu. G. Reshetnyak. Two-dimensional manifolds of bounded curvature. InGeometry, IV, volume 70 of Encyclopaedia Math. Sci., pages 3–163, 245–250. Springer, Berlin, 1993.

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