Last updated: June 27, 1999. Updates will be posted as they become available.

Notice: This material will be included in a forthcoming (summer 2000) book with the tentative title Experiencing Geometry in Euclidean, Spherical, and Hyperbolic Spaces. This new book will be an expanded and updated version of Experiencing Geometry on Plane and Sphere. This material is in draft form and may not be duplicated or quoted without the author's written permission, except for purposes of review or trying out the material with students. As always comments are welcome and will affect the final draft. Send comments to dwh2@cornell.edu.

Chapter 11

# Geometric 2-Manifolds and Coverings

There are clearly a large variety of different surfaces around in our experiential world. The study of the geometry of general surfaces is the subject of Differential Geometry. In this chapter we will study surfaces that (like cylinders and cones with the cone point removed) are locally the same (isometric) to either the plane, a sphere, or a hyperbolic plane. We study these because their geometry is simpler and closely related to the geometry we have been studying of the plane, spheres, and hyperbolic planes. In addition, the study of these surfaces will lead us to the study of the possible global shapes of our physical universe.

A geometric two-manifold is a connected space that locally is isometric to either the (Euclidean) plane, a sphere, or a hyperbolic plane. The surface of a cylinder (no top or bottom and indefinitely long) and a cone (with the cone point removed) are examples geometric two-manifolds. We use the term "manifold" here instead of "surface" because we usually think of surfaces as sitting extrinsically in 3-space. Here we want to study only the intrinsic geometry and thus any particular extrinsic embedding does not matter. Moreover, we will study some geometric two-manifolds (for example, the flat torus) which can not be (isometrically) embedded in 3-space. We ask what is the intrinsic geometric experience on geometric two-manifolds of a 2-dimensional bug. How will the bug view geodesics (intrinsically straight lines) and triangles? How can a bug on a geometric 2-manifold discover the global shape of its universe? These questions will help us as we think about how we as human beings can think about our physical universe, where we are the bugs.

This chapter will only be an introduction to these ideas. For a geometric introduction to Differential Geometry see [DG: Henderson (1998)]. For more details about geometric 2-manifolds see [DG: Weeks (1985)] and Chapter 1 of [DG: Thurston]. For the classification of (triangulated) 2-manifolds see the recent [Tp: Francis & Weeks] which contains an accessible proof due to John H. Conway.

*Problem 11.1. Geodesics on Cylinders and Cones

In Problem 2.2, we have already studied two examples of geometric 2-manifolds — cylinders and cones (without the cone point). Since these surfaces are locally isometric to the Euclidean plane, these type of geometric manifolds are called flat (or Euclidean) two-manifolds. It would be good at this point for you to review what you know from Chapter 2 about cylinder and cones.

Now we will look more closely at long geodesics that wrap around on a cylinder or cone. Several questions have arisen:

a. How many times can a geodesic on a cylinder or cone intersect itself? How are the self-intersections related to the cone angle? At what angle does the geodesic intersect itself? How can we justify this relationship?

b. How do we determine the different geodesics connecting two points? How many are there? How does it depend on the cone angle? Is there always at least one geodesic joining each pair of points? How can we justify our conjectures?

Suggestions

Here we offer the tool of covering spaces which may help you explore these questions. The method of "coverings" is so named because it utilizes layers (or sheets) that each "cover" the surface. We will first start with a cylinder because it's easier, and then move on to a cone.

n-Sheeted Coverings of a Cylinder

To understand how the method of coverings works, imagine taking a paper cylinder and cutting it axially (along a vertical generator) so that it unrolls into a plane. This is probably the way you constructed cylinders to study this problem before. The unrolled sheet (a portion of the plane) is said to be a 1-sheeted covering of the cylinder. See Figure 11.1. If you marked two points on the cylinder, A and B, as indicated in the figure, then when the cylinder is cut and unrolled into the covering, these two points become two points on the covering (which are labeled by the same letters in the figure). The two points on the covering are said to be lifts of the points on the cylinder. Figure 11.1. A 1-sheeted covering of a cylinder.

Now imagine attaching several of these "sheets" together, end-to-end. When rolled up, each sheet will go around the cylinder exactly once they will each cover the cylinder. (Rolls of toilet paper or paper towels give a rough idea of coverings of a cylinder.) Also, each sheet of the covering will have the points A and B in identical locations. You can see this (assuming the paper thickness is negligible) by rolling up the coverings and making points by sticking a sharp object through the cylinder. This means that all the A's are coverings of the same point on the cylinder and all the B's are coverings of the same point on the cylinder. We just have on the covering several representations, or lifts, of each point on the cylinder. Figure 11.2 depicts a 3-sheeted covering space for a cylinder and six geodesics joining A to B. (One of them is the most direct path from A to B and the others spiral once, twice, or three times around the cylinder in one of two directions.) Figure 11.2. A 3-sheeted covering space for a cylinder.

We could also have added more sheets to the covering on either the right or left side. You can now roll these sheets back into a cylinder and see what the geodesics look like. Remember to roll it up so that each sheet of the covering completely covers the cylinder all of the vertical lines between the coverings should lie on the same generator of the cylinder. Note that if you do this with ordinary paper, part or all of some geodesics will be hidden, even though they are all there. It may be easier to see what's happening if you use transparencies.

This method works because straightness is a local intrinsic property. Thus, lines that are straight when the coverings are laid out in a plane will still be straight when rolled into a cylinder. Remember that bending the paper does not change the intrinsic nature of the surface. Bending only changes the curvature that we see extrinsically. It is important to always look at the geodesics from the bug's point of view. The cylinder and its covering are locally isometric.

Use coverings to investigate Problem 11.1 on the cylinder. The global behavior of straight lines may be easier to see on the covering.

n-Sheeted (Branched) Coverings of a Cone

Figure 11.3 shows a 1-sheeted covering of a cone. The sheet of paper and the cone are locally isometric except at the cone point. The cone point is called a branch point of the covering. We talk about lifts of points on the cone in the same way as on the cylinder. In Figure 11.3 we depict a 1-sheeted covering of a 270° cone and label two points and their lifts. Figure 11.3. 1-sheeted covering of a 270° cone.

A 4-sheeted covering space for a cone is depicted in Figure 11.4. Each of the rays drawn from the center of the covering is a lift of a single ray on the cone. Similarly, the points marked on the covering are the lifts of the points A and B on the cone. In the covering there are four segments joining a lift of A to different lifts of B. Each of these segments is the lift of a different geodesic segment joining A to B. Figure 11.4. 4-sheeted covering space for a 89° cone.

Think about ways that the bug can use coverings as a tool to expand its exploration of surface geodesics. Also, think about ways you can use coverings to justify your observations in an intrinsic way. It is important to be precise; you don't want the bug to get lost! Count the number of ways in which you can connect two points with a straight line and relate those countings with the cone angle. Does the number of straight paths only depend on the cone angle? Look at the 450° cone and see if it is always possible to connect any two points with a straight line. Make paper models! It is not possible to get an equation that relates the cone angle to the number of geodesics joining every pair of points. However, it is possible to find a formula that works for most pairs.

Make covering spaces for cones of different size angles and refine the guesses you have already made about the numbers of self-intersections.

In studying the self-intersections of a geodesic l on a cone, it may be helpful for you to consider the ray R such that the line l is perpendicular to it. (See Figure 11.5.) Now study one lift of the geodesic l and its relationship to the lifts of the ray R. Note that the seams between individual wedges are lifts of R. Figure 11.5. Self-intersections on a cone with angle f.

Problem 11.2. Flat Torus and Klein Bottle

Flat Torus

Another example of a flat (Euclidean) two-manifold is provided by a video game that was popular a while ago. A blip on the video screen representing a ball travels in a straight line until it hits an edge of the screen. Then, the blip reappears traveling parallel to its original direction from a point at the same position on the opposite edge. Is this a representation of some surface? If so, what surface? First, imagine rolling the screen into a tube where the top and bottom edges are glued. (Figure 11.6.) This is a representation of the screen as a one-sheeted covering of the cylinder. A blip on the screen that goes off the top edge and reappears on the bottom is the lift of a point on the cylinder which travels around the cylinder crossing the line which corresponds to the joining of the top and bottom of the screen. Figure 11.6. One-sheeted covering of cylinder and flat torus.

Now, let us further imagine that the cylinder can be stretched and bent so that we can glue the two ends to make a torus. Now the screen represents a one-sheeted covering of the torus. If the blip goes off on one side and comes back on the other at the same height, this represents the lift of a point moving around the torus and crossing the circle which corresponds to the place where the two ends of the cylinder are joined. The possible motions of a point on the torus are represented by the motions on the video screen! Figure 11.7. Non-flat torus.

But the torus pictured in Figure 11.7 is not a geometric 2-manifold because the original flat (Euclidean) geometry has been distorted and is not exactly either spherical or hyperbolic. You can't make a model in 3-space of a torus from a flat piece of paper without distorting it, but you can in 4-space! Such a torus is called a flat torus. It is best not to call this a "surface", because there is no way to realize it isometrically in 3-space and it is not the surface of anything. But the question of whether or not you can make an isometric model in 3-space is not important — the point is that the gluings in Figure 11.6 intrinsically define a flat 2-manifold.

a. Show that the flat torus is locally isometric to the plane and thus, is a geometric two-manifold, in particular, a flat (Euclidean) two-manifold

[Hint: Note that each point on the interior of an edge of the screen is the lift of a point which has another lift on the opposite edge. Thus, a lift of a neighborhood of that is in two pieces (one near each of the two opposite edges). What happens at the four corners of the computer screen (which are lifts of the same point?]

The torus in Figure 11.7 and the flat torus are related in that there is a continuous one-to-one mapping from either to the other. We say that they are homeomorphic, or topologically equivalent; and a homeomorphism is a mapping which is continuous and one-to-one and whose inverse is also continuous. We can further express this situation by saying that the torus in Figure 11.7 and the flat torus are both topological tori.

There is another representation of the flat torus based on a hexagon. Start with a regular hexagon in the plane and glue opposite sides as indicated in Figure 11.8. Figure 11.8. Flat torus from a hexagon.

b. Show that gluing the edges of the hexagon as in Figure 11.8 forms a flat 2-manifold which is homeomorphic to the flat torus.

*Flat Klein bottle (This Klein bottle material may be skipped)

Now we describe a related geometric two-manifold which is traditionally called a flat Klein bottle. Imagine the same video screen, again with a traveling blip representing a ball which travels in a straight line until it hits an edge of the screen. When it hits the top edge then the blip proceeds exactly the same as for the flat torus (traveling parallel to its original direction from a point at the same position on the opposite edge). However, when the blip hits a vertical edge of the screen it reappears on the opposite edge but in the diametrically opposite position and travels in a direction that has slope which is the negative of the original slope. (See Figure 11.9.) As before, imagine rolling the screen into a tube where the top and bottom edges are joined. This is again a representation of the screen as a one-sheeted covering of the cylinder. Figure 11.9. One-sheeted covering of a flat Klein bottle.

A blip on the screen that goes off the top edge and reappears on the bottom is the lift of a point on the cylinder which travels around the cylinder crossing the line which corresponds to the joining of the top and bottom of the screen.

Now, let us further imagine that the cylinder can be stretched and bent so that we can join the two ends to make a topological Klein bottle which is not a geometric 2-manifold. See Figure 11.10. Now the screen represents a one-sheeted covering of the Klein bottle. The possible motions of a point on the Klein bottle are represented by the motions on the video screen! Figure 11.10. Topological Klein bottle.

You also can't make an isometric model in 3-space of a flat Klein bottle without distorting it and having self-intersections. But the gluings in Figure 11.9 define intrinsically a flat 2-manifold.

*c. Show that the flat Klein bottle is locally isometric to the plane and thus is a geometric two-manifold, in particular, a flat (Euclidean) two-manifold.

[Hint: Note that the four corners of the video screen are lifts of the same point and that a neighborhood of this point has 360° — that is, 90° from each of the four corners.]

It can be shown that: (For a detailed discussion, see [DG: Thurston (1997)], pages 25-18. For a more elementary discussion see [DG: Weeks (1985)], Chapters 4 and 11.)

Theorem. Flat tori and flat Klein bottles are the only flat (Euclidean) 2-manifolds which are finite and geodesically complete (every geodesic can be extended indefinitely).

Note that if a finite cylinder is not geodesically complete and if it is extended indefinitely then it is geodesically complete but then is not finite. A cone with the cone point is not a flat manifold at the cone point; with the cone point removed the cone is not geodesically complete.

Note that we get a flat tori for each size rectangle in the plane. These flat tori are different geometrically because there are different distances around the tori. However, topologically they are all the same as (homeomorphic to) the surface of a donut.

Note that if you move a right hand glove (which we stylize by ) around the flat torus it will always stay right-handed; however, if you move it around the flat Klein bottle horizontally it will become left-handed. See Figure 11.11. We describe these phenomena by saying that the flat torus is orientable and that a Klein bottle is non-orientable. Figure 11.11. Orientable and non-orientable.

*Problem 11.3. Universal Covering of Flat 2-Manifold

a. On a flat torus or flat Klein bottle, how do we determine the different geodesics connecting two points? How many are there? How can we justify our conjectures?

[Hint: Look at straight lines in the universal coverings introduced below.]

b. Show that some geodesics on the flat torus or flat Klein bottle are closed curves (in the sense that they come back and continue along themselves like great circles), though possibly self-intersecting. How can you find them?

[Hint: Look in the universal coverings introduced below.]

c. Show that there are geodesics on the flat torus and flat Klein bottle which never come back and continue along themselves, but yet come arbitrarily close to every point in the manifold (such curves are said to be dense).

[Hint: Look at the slopes of the geodesics found in Part b.]

The geodesics found in Part c can be shown to come arbitrarily close to every point on the manifold. Such curves are said to be dense in the manifold.

Suggestions

I suggest that you use coverings just as you did for cones and cylinders. The difference is that in this case the sheets of the coverings extend in two directions. See Figure 11.12 for a covering of the flat torus. If this covering is continued indefinitely in all directions then the whole plane covers the flat torus with each point in the torus having infinitely many lifts. When a covering is the whole of either the (Euclidean) plane, a sphere, or a hyperbolic plane it is called the universal covering. See Figure 11.13 for a universal covering of a flat Klein bottle. These coverings are called "universal" because there are no coverings of the plane, spheres, or hyperbolic planes which have more than one sheet — see the next section for a discussion of this for a sphere. Figure 11.12. Universal covering of a flat torus. Figure 11.13. Universal covering of the flat Klein bottle.

Problem 11.4. Spherical 2-Manifolds

Start by considering another version of the video screen as depicted in Figure 11.14. Imagine the same video screen, again with a traveling blip representing a ball which travels in a straight line until it hits an edge of the screen. Now when the blip reaches any edge of the screen it reappears on the opposite edge but in the diametrically opposite position and travels in a direction that has slope which is the negative of the original slope. (See Figure 11.14.) Figure 11.14. This is not a geometric 2-manifold.

a. Show that the situation in Figure 11.14 does not represent a geometric 2-manifold because the corners represent two cone point with cone angle 180°.

[Hint: Cut around the corners marked G and tape together the edges as indicated.]

The gluings indicated in Figure 11.14 fail to produce a geometric 2-manifold because the interior angles are only 90°. Thus, it seems that we might get a geometric manifold if the interior angles were 180°. Thus, we need a quadrilateral with equal opposite sides and interior angles with 180°. There is no such quadrilateral in the plane; however, on the sphere there IS such a quadrilateral! See Figure 11.15. Figure 11.15. Gluings on a hemisphere producing a projective plane.

What we have in Figure 11.15 is a hemisphere with each point of the bounding equator being glued to its antipode, in this way one gets what is called the (real) projective plane, often denoted RP2.

b. Show that a projective plane is a spherical 2-manifold and that it is non-orientable.

[Hint: Examine the neighborhood of a point of the projective plane that comes from the bounding equator.]

c. What are the geodesics on a projective plane?

If you cut out and remove a lune from a sphere and then join together the two edges of the lune you what can reasonably be called a spherical cone. See Figure 11.16. Note that by removing the appropriate lune these spherical cones have a shape similar to an American football. Figure 11.16. A spherical cone.

By pasting together several (equal radius) spheres with the same lune removed you can get multiple sheeted branched coverings of a spherical cone. A spherical cone with the two cone points removed is a finite spherical 2-manifold but (as with ordinary cones) it is not geodesically complete.

d. Show that the spherical cones as described above are orientable spherical 2-manifolds, if you remove the two cone points.

e. Identify the geodesics on a spherical cone with cone angle 180° (that is, you remove from the sphere a lune with angle 360° – 180° = 180°). What happens with other cone angles?

[Hint: Look at the great circles in the sphere minus the lune before its edges are joined to produce the spherical cone.]

It can be shown that:

Theorem. Spheres and projective planes are the only spherical 2-manifolds which are finite and geodesically complete (every geodesic can be extended indefinitely).

For a detailed discussion, see [DG: Thurston (1997)], pages 25-18. For a more elementary discussion see [DG: Weeks (1985)], Chapters 4 and 11.

*Coverings of the Sphere

There is no way to construct a covering of a sphere that has more than one sheet unless the covering has some "branch points." A branch point on a covering is a point such that every neighborhood (no matter how small) surrounding the point contains at least two lifts of some point. In any covering of a cone with more than one sheet, the lift of the cone point is a branch point as you can see in Figure 11.17. Figure 11.17. Covering space of a cone has branch points.

Notice that the coverings of a cylinder and a flat torus have no branch points. For a sphere the matter is very different any covering of a sphere will have a branch point. You can see this if you try to construct a cover by slitting two spheres as depicted in Figure 11.18 and then sticking the two together along the slit. The ends of the slit would become branch points. This topic may be explored further in textbooks on geometric or algebraic topology. Figure 11.18. Covering space of a sphere has branch points.

In fact, any surface that has no (non-branched) coverings and which is bounded and without an edge can be continuously deformed (without tearing) into a round sphere. The surfaces of closed boxes and of footballs are two examples. A torus is bounded and without an edge, but it cannot be deformed into a sphere. A cylinder also cannot be deformed into a sphere, and a cylinder either has an edge or (if we imagine it as extending indefinitely) it is unbounded.

A 3-dimensional analog of this situation arises from a famous, long-unsolved problem called the Poincaré Conjecture. The analog of a surface is called a 3-dimensional manifold, a space which is locally like Euclidean 3-space, (in the same sense that a surface is locally like the plane). The 3-sphere, which we will study in Chapter 12, is a 3-dimensional manifold. Also, our physical 3-dimensional universe may not be Euclidean; but, certainly, it is locally like Euclidean 3-space. Thus, is a 3-dimensional manifold. Poincaré (1854-1912, French) conjectured that any 3-dimensional manifold which has no (non-branched) coverings and which is bounded and without boundary must be homeomorphic to a 3-dimensional sphere S3. For the past 80 years, numerous mathematicians have tried to decide whether Poincaré's conjecture is true or not. So far, no one has succeeded. See [SE: Hilbert] and [DG: Weeks] for more discussion of 3-dimensional manifolds and the 3-dimensional sphere.

Problem 11.5. Hyperbolic Manifolds

Now is it possible to make a two-holed torus (sometimes called an anchor ring, or the surface of a two-holed donut) into a geometric 2-manifold? See Figure 11.19. Figure 11.19. Two-holed torus.

Note that as it is pictured in Figure 11.19 the two-holed torus is definitely NOT a geometric 2-manifold because the locally geometry is not the same at every point — near the point A the surface appears to have a sphere-like geometry and near the point B the surface appears to have a hyperbolic-like geometry. But can we distort the geometry so that the surface is a geometric 2-manifold? Figure 11.20. Cutting a two-holed torus.

Imagine cutting the two-holed torus along the four loops emanating from the point P, as indicated in Figure 11.20. You will get a distorted octagon with 45° (= 360°/8) interior angles at each vertex. This distorted octagon is topologically equivalent to a regular planar octagon. Walking around the point P we find the gluings as indicated in Figure 11.20. Be sure you understand how the gluings on the octagon where determined from the loops on the two-holed torus. If you glue the edges of the regular octagon together as indicated in Figure 11.20, we will get a version of the two-holed torus that is geometrically the plane except at the (one) vertex. (Why? What will a neighborhood of the vertex look like?) In the plane all octagons have the same interior angle sum. But, in the hyperbolic plane, regular octagons have different angles. In fact, we can find such an octagon in the hyperbolic plane with 45° interior angles. (See Figure 11.21). If we glue the edges of this octagon as indicated then we will get a hyperbolic 2-manifold. Figure 11.21. Hyperbolic octagon with 45° angles.

To see that there is such an octagon, imagine placing a small (regular) octagon on the hyperbolic plane. Since the octagon is small its interior angles must be very close to the interior angles of an octagon in the (Euclidean) plane. Since the exterior angle of an planar octagon must be 360°/8, the interior angle must be 180° – (360°/8) = 135°. Now let the small octagon grow, keeping it always regular. From Problem 7.2 (remembering that an octagon can be divided into triangles as in Figure 11.21), we conclude that as the interior angles of the octagon will decrease in size until if we let the vertices go to infinity the angles would decrease to zero. Some where in between the interior angles will be the desired 45°.

a. What is the area of the hyperbolic octagon with 45° interior angles?

b. Why is the two-holed torus obtained from a hyperbolic octagon with 45° interior angles is hyperbolic 2-manifold? What is its area?

It is not easy to determine the geodesics on a hyperbolic 2-manifold. Some discussion about these issues is contained in William Thurston's Three-Dimensional Geometry and Topology, Volume 1 [DG: Thurston (1997)] and will presumably be continued in Volume 2.

There are other ways of making a two-holed torus into a geometric 2-manifold but always it is a hyperbolic 2-manifold. However, there are many different hyperbolic structures for a two-holed donut; for example, look in Figure 11.22 for a different way to represent the two-holed torus this time as the gluing of the boundary of a dodecagon with 90° interior angles. Figure 11.22. Cutting two-holed torus into a dodecagon.

c. Follow the steps above to check that Figure 11.22 leads to the representation of a two-holed torus as a dodecagon (with 90° interior angles) with gluing on the boundary and thus to a hyperbolic 2-manifold. What is its area?

You should have found the same area in Parts b and c. In fact, we (you!) will show in the next problem that any hyperbolic geometric manifold structure on a 2-holed torus has the same area.

Problem 11.6. Area, Euler Number and Gauss-Bonnet

Now is a good time to go back and review the material on area and holonomy in Problems 7.1 - 7.4a. Recall that you showed that the area of any polygon is:

K Area(G) = [ S bi - (n - 1)p ],

where K is the Gaussian curvature equal to 1/r2 for a sphere and -1/r2 for a hyperbolic plane of radius r, and S bi is the sum of the interior angles, and n is the number of edges.

We will now use those results to study the area of geometric manifolds. All of the geometric manifolds that we have described above have cell-divisions. A 0-cell is a point which we usually call a vertex. A 1-cell is a straight line segment which we usually call an edge (the edge need not be straight but in most of our applications it will be). A 2-cell is a polygon which is usually called a face. We say that a geometric manifold has a cell-division if it is divided into cells so that every edge has its boundary consisting of vertices and every face has it boundary divided into edges and vertices and two cells only intersect on their boundaries. We call the cell-division a geodesic cell-division if all the edges are geodesic segments, and thus the faces are polygons. For example, in Figure 11.22 we have a two-holed torus divided into one face, six edges, and three vertices. Note that some edges have only one vertex and thus form a loop (circle). If we make this two-holed torus into a hyperbolic manifold (by using a regular hyperbolic dodecagon with 90° angles) then the cell-division is a geodesic cell-division.

If we have a geodesic cell-division of geometric manifold M, then we can calculate the area of M as follows:

K Area(M) = {area(1st face)} + {area(2nd face)} + ... + {area(k-th face)}
= {Sbi (1st face) - (n1-2)p} + ... + {Sbi (k-th face) - (nk-2)p}
= {Sbi (1st face) +...+ Sbi (k-th face)} - {(n1-2)p +...+ (nk-2)p}
= {sum of all the angles} - (n1+n2+...+nk)p + (2+2+...+2)p

Fill in the steps to prove:

a. If a geometric manifold M has a geodesic cell-division then

K area (M) = 2p ( v - e + f ),

where the cell-division has v vertices and e edges and f faces. Check that this agrees with your results in Problems 11.2, and 11.4.

The quantity ( v - e + f ) is called the Euler number (or sometimes the Euler characteristic). This quantity is named after the mathematician Leonhard Euler (1707-1783) who was born and educated in Basel but worked in St. Petersburg and Berlin. It follows directly from Part a and what we know about area and curvature that:

b. The Euler number of any geodesic cell-division of a sphere must be 2. The Euler number of any geodesic cell-division of a projective plane must be 1. The Euler number of any geodesic cell-division of a flat (Euclidean) 2-manifold must be equal to 0. The Euler number of any geodesic cell-division of a hyperbolic 2-manifold must be negative.

We see from Part b that in the cases of a sphere, a projective plane, or a flat 2-manifold, the Euler number does not depend on the specific cell-division (with geodesic edges). So, we can talk about the Euler number of the sphere (= 2) and the Euler number of the torus or Klein bottle (= 0). We also saw above that the two different cell-divisions that were given of the two-holed torus have the same area and thus the same Euler number. Can we prove that for every hyperbolic 2-manifold the Euler number (and therefore the area) depends only on the topology and not on the particular cell-division? In fact:

Theorem. The Euler number of any cell-division of a 2-manifold depends only on the topology of the manifold and not on the specific cell-division. Furthermore, two 2-manifolds are homeomorphic if and only if they have the same Euler number and are either both orientable or both non-orientable.

Proofs of this result are somewhat fussy and involves much of the foundational results of topology which only developed in Twentieth Century. See Imre Lakatos's Proofs and Refutations [Ph: Lakatos] for an accessible and interesting account of the long and complicated history and philosophy of the Euler number. Lakatos describes an imaginary class discussion about the Euler number in which the tortuous route students take towards a proof mirrors the actual route that mathematicians took. Other proofs are with different additional assumptions. For example, [DG: Thurston] (Propositions 1.3.10 and 1.3.12) gives an accessible proof assuming that the reader has some familiarity with vector fields on differentiable manifolds. In Sections 2.4 and 2.5 of [Tp: Blackett] there is a combinatorial-based proof that assumes the topological 2-manifold has some cell-division.

Triangles on Geometric Manifolds

Clearly, if on a flat (Euclidean) 2-manifold a triangle is contained in a region that is isometric to the plane, then the triangle is a planar triangle and has all the properties of a triangle in the plane. The same can be said about triangles in spherical and hyperbolic 2-manifolds. In fact it can be shown (see any topology text that deals with covering spaces):

Theorem. If D is a triangle in a Euclidean [spherical, hyperbolic] 2-manifold, M, such that D can be shrunk to a point in the interior of D, then D (and its interior) can be lifted to the plane [sphere, hyperbolic plane] which is the universal covering space of M; and thus, D has all the same properties of a triangle in the plane [sphere, hyperbolic plane].

It is natural to be uncomfortable using covering spaces, but covering spaces are a helpful tool for thinking intrinsically. Some triangles, even though they look strange extrinsically, will look like reasonable triangles for the bug. In Figure 11.23 we give an example of an extrinsically strange triangle which intersects itself but which can be considered a normal triangle from an intrinsic point of view. In fact, it is a planar triangle. Such triangles have all the properties of plane triangles including SAS and ASA. Unroll to a 3-sheeted cover, and ... Figure 11.23. Think intrinsically. This is a triangle!

Problem 11.7. Can The Bug Tell Which Manifold?

Our physical universe is apparently a geometric 3-manifold. In Chapter 20 we will explore ways in which we (human beings) may be able to determine the global shape and size of our physical universe. But, first, we look at the situation of a 2-dimensional bug on a geometric 2-manifold in order to get some help for our 3-dimensional question.

a. Suppose a 2-dimensional bug lives on a geometric 2-manifold M and suppose that M is the bug's whole universe. How can the bug determine intrinsically which 2-manifold it is on? For this part you may imagine that the bug can crawl over the whole manifold and leave markers and make measurements.

b. Suppose that the bug in Part a can only travel in a very small region of the manifold (so small that all triangles in the region are indistinguishable from planar triangles), but suppose that the bug can see for very long distances. Can the bug still determine which geometric 2-manifold it is on?