edge. However, if the angles of the corners that come together at a point do not add up to 360◦, then the surface has either a cone point or a saddle-point,
depending on whether the angle sum is less than or more than 360◦. As we
saw in Chapter 1 (Exercises 1.3.5 and Checkpoint 1.3.7), in either case, we will not have a homogeneous geometry; a two-dimensional bug would be able to distinguish (with triangles or circles) a cone point from a at point from a saddle point.
To smooth out such cone points or saddle points we change the angles at the corners so that the angles do add up to 360◦. We now have the means for
doing this. If we need to shrink the corner angles, we can put the polygon in the hyperbolic plane. If we need to expand the angles, we can put the polygon in the projective plane.
Example 7.6.1: C3 admits hyperbolic geometry.
The standard polygonal representation of C3 = P2#P2#P2, is a
hexagon having boundary label a1a1a2a2a3a3, as in Figure 7.6.2. All
six corners of the hexagon come together at a single point. In the Euclidean plane, a regular hexagon has corner angles equal to 120◦. To
avoid a saddle point when joining the six corners together, shrink the corner angles to 60◦. A tiny copy of a regular hexagon in the hyperbolic
plane will have corner angles just under 120◦. If the hexagon grows so
that its vertices approach ideal points, its corner angles will approach 0◦. At some point, then, the interior angles will be 60◦ on the nose.
(We may construct this precise hexagon as well. See Exercise 7.6.4.) If C3is built from this hexagon living in the hyperbolic plane, the surface
inherits the geometry of the space in which it nds itself; that is, the surface C3 admits hyperbolic geometry, a nice homogeneous, isotropic
and metric geometry.
a2 a2 a3 a3 a1 a1 a2 a2 a3 a3 a1 a1
Figure 7.6.2: C3 admits hyperbolic geometry.
Example 7.6.3: An elliptic polygonal surface.
Revisiting Example 1.3.9, consider the hexagon with boundary label abcabc. The six corners of this polygonal surface come together in groups of two. These corners create cone points because the angle sum of the two corners coming together is less than 360◦ in the Euclidean
plane. We can avoid cone points by putting the hexagon in P2. A
small regular hexagon in the projective plane will have corner angles just slightly greater than 120◦, but we need each corner angle to expand
covers the entire projective plane. In fact, the surface of Example 1.3.9 is the projective plane, and it admits elliptic geometry.
Theorem 7.6.4. Each surface admits one homogeneous, isotropic, and metric geometry. In particular, the sphere (H0) and projective plane (C1) admit
elliptic geometry. The torus (H1) and Klein bottle (C2) admit Euclidean
geometry. All handlebody surfacesHgwithg≥2and all cross-cap surfacesCg
withg≥3 admit hyperbolic geometry.
The following section oers a more formal discussion of how any surface admits one of our three geometries but we present an intuitive argument here. The sphere and projective plane admit elliptic geometry by construction: The space in elliptic geometry is the projective plane, and via stereographic projection, this is the geometry onS2.
The torus and Klein bottle are built from regular 4-gons (squares) whose edges are identied in such a way that all 4 corners come together at a point. In each case, if we place the square in the Euclidean plane all corner angles are π/2, so the sum of the angles is 2π, and our surfaces admit Euclidean geometry. Each handlebody surfaces Hg forg≥2and each cross-cap surfacesCg for
g ≥3 can be built from a regularn-gon where n ≥ 6. Again, all n corners come together at a single point. A regularn-gon in the Euclidean plane has interior angle(n−2)π/nradians, so the corner angles sum to(n−2)πradians. This angle sum exceeds2πradians sincen≥6. Placing a small version of this
n-gon in the hyperbolic plane, the corner angle sum will be very nearly equal to (n−2)π radians and will exceed 2π radians, but as we expand the n-gon the corners approach ideal points and the corner angles sum will approach 0 radians. Thus, at some point the angle sum will equal 2π radians on the nose, and the polygonal surface built from this precisen-gon admits hyperbolic geometry. Exercise 7.6.4 works through how to construct this precisen-gon.
Of course, one need not build a surface from a regular polygon. For instance, the torus can be built from any rectangle in the Euclidean plane and it will inherit Euclidean geometry. So while the type of geometry our surface admits is determined, we have some exibility where certain geometric measurements are concerned. For instance, there is no restriction on the total area of the torus, and the rectangle on which it is formed can have arbitrary length and width dimensions. These dimensions would have a simple, tangible meaning to a two-dimensional bug living in the surface (and might be experimentally determined). Each dimension corresponds to the length of a geodesic path that would return the bug to its starting point. A closed geodesic path in a surface is a path that follows along a straight line (in the underlying geometry) that starts and ends at the same point. Figure 7.6.5 shows three closed geodesic paths, all starting and ending at a point near a bug's house. The length of one path equals the width of the rectangle, the length of another equals the length of the rectangle, and the third follows a path that is longer than the rst two.
7.6. Geometry of Surfaces 165
Figure 7.6.5: A at torus has many closed geodesic paths. The length of the shortest closed geodesic path equals the length of the short side of the rectangle on which the torus is modeled.
Even in hyperbolic surfaces, where the area of the surface is xed (for a given curvature) by the Gauss-Bonnet formula, which we prove shortly, there is freedom in determining the length of closed geodesic paths.
Example 7.6.6: Building hyperbolic surfaces from pants. If we make three slices in the two-holed torus we obtain two pairs of pants, as indicated in the following gure.
c
1c
2d
1d
2e
1e
2c
2c
1d
2d
1d
2d
1c
2c
1e
2e
1e
2e
1We label our cuts so that we can stitch up our surface later. Match theci,di, andeiedges to recover the two-holed torus. Any pair of pants
can be cut into two hexagons by cutting along the three vertical seams in the pants. It follows that the two-holed torus can be constructed from four hexagons, with edges identied in pairs as indicated below.
c2 c1 d2 d1 d2 d1 a1 a3 a2 a3 c1 a1 d1 a2 d1 a3 c2 a1 d2 a2 d2 c2 c1 e2 e1 e2 e1 b1 b3 b2 b3 c1 b1 e1 b2 e1 b3 c2 b1 e2 b2 e2
The four hexagons represent a cell division ofH2having 6 vertices,
12 edges, and 4 faces. In the edge identication, corners come together in groups of 4, so we need each corner angle to equal 90◦ in order to
endow it with a homogeneous geometry. We know we can do this in the hyperbolic plane. Moreover, according to Theorem 5.4.19, there is freedom in choosing the dimensions of the hexagons. That is, for each triple of real numbers(a, b, c)there exists a right-angled hexagon inD
with alternate lengthsa, b,andc. So, there exists a two-holed torus for each combination of six seam lengths (a1, a2, a3, b1, b2,andb3).
A surface that admits one of our three geometries will have constant curvature. The reader might have already noticed that the sign of the curvature will equal the sign of the surface's Euler characteristic. Of course the magnitude of the curvature (ifk6= 0)can vary if we place a polygonal surface in a scaled
version ofP2orD. That is, while the type of homogeneous geometry a surface
admits is determined by its Euler characteristic (which is determined by its shape), the curvature scale can vary if k 6= 0. By changing the radius of a
sphere, we change its curvature (though it always remains positive). Similarly, the surfaceC3 in Figure 7.6.2 has constant curvature -1 if it is placed in the
hyperbolic plane of Chapter 5. However, it can just as easily nd itself in the hyperbolic plane with curvature k = −8. Recall, the hyperbolic plane with
curvaturek <0 is modeled on the open disk inCcentered at the origin with
radius1/p|k|. Placing the hexagonal representation of C3 into this space so
that its corner angle sum is still 2πproduces a surface with constant curvature k.
We have nally arrived at the elegant relationship between a surface's curvaturek, its area, and its Euler characteristic. This relationship crystalizes the interaction between the topology and geometry of surfaces.
Theorem 7.6.7 Gauss-Bonnet. The area of a surface with constant curva- ture kand Euler characteristic χis given by the formula kA= 2πχ.
Proof. The sphere with constant curvaturekhas radius equal to1/√k, and area equal to 4π/k. Since the sphere has Euler characteristic 2, the Gauss- Bonnet formula holds in this case. The projective planeP2 with curvaturek
has area equal to2π/k, and Euler characteristic equal to 1, so the result holds in this case as well. The torus and the Klein bottle each havek= 0andχ= 0,