Euler Characteristic of Two-Dimensional Manifolds

Transcription

1 Euler Characteristic of Two-Dimensional Manifolds M. Hafiz Khusyairi August 2008 In this work we will discuss an important notion from topology, namely Euler Characteristic and we will discuss several definitions of this notion. These several definitions vary from topological, geometrical, analytical, and even combinatorial approach. Hence, we can see this concept as a bridge connecting several areas of mathematics. We will also make these definitions precise by assuring the their well-definedness. 1 Definition by Triangulation The definition which is most common in literature, and usually the easiest to be computed is the definition by triangulation. Originally defined by Leonhard Euler for convex polyhedra by the means of triangulation, the concept can be generalized for two-dimensional compact orientable manifolds in R 3. Troughout this writing M would be such a manifold together with a Riemannian metric, and also assume that M can be decomposed into a finite number of geodesic nonoverlapping triangles. Here, nonoverlapping means for two triangles in the triangulation their intersection must be either empty, a common vertex or a common edge. The definition of the Euler Characteristic is the following. Definition 1.1. Let M be the above mentioned manifold, and assume that it can be triangulated as above. Let V, E, and F be the total number of vertices, edges and faces of the triangulation respectively. Then the Euler Characteristic of M is defined as χ(x) = V E + F. In order that the Euler Characteristic is well-defined then such decomposition should always exists and the Euler Characteristic should be independent of the triangulation. We will use the fact that the triangulation exists without giving the proof. The second fact would be discussed in the last section. Example 1.2. We will use the fact that Euler Characteristic is topologically invariant number to compute the Euler Characteristics of sphere and torus. Topologically invariant means if M and N are homeomorphic, then χ(m) = χ(n). We know that boundary of a cube is homeomorphic to a sphere. One possible triangulation of cube can be seen in Figure 1. The Euler Characteristic of the 1

2 Figure 1: The triangulation of cube s boundary Figure 2: A manifold homeomorphic to torus cube s boundary can be easily computed, and the number turns out to be 2. As for torus, we will make use the manifold on Figure 2 which is homeomorphic to torus. One simplest triangulation can be defined as follows. Decomposed the manifold into four manifolds (which is homeomorphic to boundary of a cube) as in Figure 3. The Euler Characteristic of this manifold is the sum of the each decomposition s Euler Characteristic (8) subtracted by the number of common vertices (16) and the number of faces that would be erased in gluing the manifolds together (16) and added by the number of edges that would be erased in the gluing (40). Hence the Euler Characteristic of the manifold is zero and so is the torus. 2 Definition by Indices and Angle Function Now, by making use of vector field we will discuss another definition of Euler Characteristic,. Unlike the previous definition, we will need the manifolds associated to be differentiable. Of course we expect the two definitions to agree if Figure 3: The decomposition into 4 cube s-boundary-like manifolds 2

3 they are both defined. 2.1 Moving Frames We begin by introducing the so called moving frames. Definition 2.1. Let U R n be an open set and let {e i }, i = 1,..., n be n differentiable vector fields on U such that for each e i, e j = δ ij. Here, δ ij is the Kronecker delta symbol. The set {e i }, i = 1,..., n is called a moving frame on U. For a moving frame {e i }, i = 1,..., n on U R n we can define Definition 2.2. Let {ω i }, i = 1,..., n be a set of differential forms such that ω i (e j ) = δ ij. The set {ω i }, i = 1,..., n is called coframe associated to {e i }. Definition 2.3. For each p R n (point in R n ) and v R n (vector in tangent space of R n ), we know that (de i ) p (v) = (de i ) p (v), e j (p) e j (p). Define ω ij as the differential forms that map v to (de i ) p (v), e j (p). The set {ω ij }, i, j = 1,..., n is called connection forms of U in the moving frame {e i }. Remark 2.4. Differentiating both sides of equation e i, e j = δ ij gives that is ω ij = ω ji 0 = de i, e j = δ ij + e i, de j = δ ij = ω ij + ω ji, Let x : U R n be the inclusion map. Throughout this work, x will denotes inclusion, immersion or embedding map, and will often be omitted for simplicity. Then, we know that dx = n ω ie i. From the fact that d 2 x = 0, we will derive the necessary condition for a moving frame, that is the so called structure equations of the moving frame. Observe that, since ω i is a 1-form, then 0 = d(dx) = = = = dω i e i ω i de i dω i e i dω i e i (dω i e i ω j de j ω j ω ji e i ω j ω ji )e i, 3

4 which gives the first structure equations dω i = ω k ω ki. k=1 By using the relationship d(de k ) = 0, we have 0 = d(de k ) = = = dω ki e i ω ki de i dω ki e i dω ki e i = ω kj de j ω kj (dω ki e i that gives us the second structure equations dω ki = ω ji e i ω kj ω ji )e i, ω km ω mi. m=1 Since we are going to make use of vector field which is defined on the tangent space of M then it is natural to define the following. Definition 2.5. Let M be m-dimensional manifold x : M R n+k, and U is a neighborhood of a point p M. Let V R n+k such that V M = x(u) (here we omitted x from x(m), later in this work it will be common to omit x when it is not confusing). If there exists a moving frame {e 1,..., e n, e n+1,..., e n+k }, such that when restricted to x(u), the vectors e 1,..., e n are tangent to x(u), then such a moving frame is called an adapted frame. Let consider the case when M is two-dimensional manifolds and n + k = 3. Let e 1, e 2, e 3 be and adapted moving frame of x(u M). Note that ω 3 (v) = ω 3 (a 1 e 1 + a 2 e 2 ) = 0, where v = a 1 e 1 + a 2 e 2 T p M. Since ω 3 = ω ii = 0, the structure equations become dω 1 = ω 2 ω 21 dω 2 = ω 1 ω 12 dω 3 = ω 1 ω 13 dω 12 = ω 13 ω 32 dω 13 = ω 12 ω 23 dω 23 = ω 21 ω 13. 4

5 As we wanted the will-be-defined Euler Characteristic to be topologically invariant, we then want to have the Euler Characteristic to be an intrinsic geometry notion of the manifold. That is, the Euler Characteristic is independent of the parametrization or the embedding. This way, the Euler Characteristic should be topologically invariant. We will define the Euler Characteristic which only depend on ω 1, ω 2, and ω 12. Now, we will explain why they (ω 1, ω 2, ω 12 ) constitute the intrinsic geometry. For p M, we can choose U neighborhood of p such that we can define the moving frame e 1, e 2. From this moving frame, we can uniquely determine ω 1, ω 2, the coframe associated to e 1, e 2. ω 12 can also be uniquely determined, which is the content of the following theorem (Theorem of Levi-Civitta). Theorem 2.6. Let M be a Riemannian two-dimensional manifold. Let U M be an open set where the moving frame {e 1, e 2 } is defined, and let {ω 1, ω 2 } be the associated coframe. Then there exists a unique 1-form ω 12 = ω 21 such that dω 1 = ω 12 ω 2 and dω 2 = ω 21 ω Angle Function and Indices Suppose that we have a change of frame, and the angle between old and new frames is a differentiable function. First we want to elaborate the notion of change-of-frame. Let {e 1, e 2 }, and {ē 1, ē 2 } be two moving frames on U. Then we know that ē 1 = fe 1 + ge 2, for some f, g differentiable functions. We have 1 = ē 1, ē 1 = f 2 + g 2. Since ē 1, ē 2 = 0, then ē 2 = ge 1 + fe 2 (the two frames have same orientations) or ē 2 = ge 1 fe 2 (those two frames orientations are opposite). From now, we will restrict our attention only to orientable surfaces, therefore we can consider only the first case. Observe that e 1 = fē 1 gē 2 and e 2 = gē 1 +fē 2. So, we also have ω 1 = f ω 1 g ω 2 and ω 2 = g ω 1 + f ω 2. Differentiating the first equation we obtain dω 1 = df ω 1 + fd ω 1 dg ω 2 gd ω 2, simplifying the terms by using structure equations and some equations above, we obtain dω 1 = ω 12 ω 2 τ ω 2 = ( ω 12 τ) ω 2, where τ = fdg gdf. Similarly, we have We have the following lemma. dω 2 = ( ω 12 τ) ω 1. Lemma 2.7. Let U R n and let {ω i }, i = 1,..., n be linearly independent differential 1-forms in U. If there exists a set of forms {ω ij }, i, j = 1,..., n that 5

6 satisfy the conditions: then such a set is unique. ω ij = ω ji, and dω j = ω k ω kj, From the lemma above, ω 12 = ω 12 τ. Now, we will observe that τ is actually is the differential of the angle function between e 1 and ē 1 along the curve. This is the content of the following lemma. Lemma 2.8. Let p U M be a point and let γ : I U be a curve such that γ(t 0 ) = p. Let φ 0 = angle(e 1 (p), ē 1 (p)). Then, φ(t) = t is a differentiable function such that k=1 t 0 (f dg dt g df dt )dt + φ 0 cos φ(t) = f, sin φ(t) = g, φ(t 0 ) = φ 0, dφ = γ τ. Consider X a differentiable vector field on M. We called points p where X(p) = 0 as the singularity points of X. Let restrict our attention to the case where M compact and all the singularity of X is isolated. Suppose that there are infinitely many points of these singularity. Then, we can define a sequence of isolated singularity points, since M is compact then this sequence has a converging subsequence which limit is in M. But then this limit is no more an isolated singularity which is a contradiction. Hence, the number of singularities are finite. We now gives the definition of Euler characteristic. Definition 2.9. Let {p i }, i = 1,..., k denotes the set of all singularities of X on M. For each p i, consider the neighborhood U i small enough so that there is only one singularity inside U i. For each U i {p i }, we define a moving frame {ē 1, ē 2 }, where ē 1 = X/ X and ē 2 is orthonormal to ē 1 such that {ē 1, ē 2 } has the same orientation as M. Let, {e 1, e 2 } is a moving frame with the same orientation as M, then we know that ω 12 ω 12 = τ. Now consider a simple closed curve C that bounds U i with the orientation of boundary of U i. Let I i satisfies τ = dφ = 2πI i, C C then if we consider the endpoint of C as t 0 then we know after following the curve C the angle at t 0 will come back to where it starts. Then I i is an integer, we called this number as the index of X at p. We define the number χ X (M) = k I i as the Euler characteristic of M. The well-definedness is just the direct implication of the three following lemmas. 6

7 Figure 4: The calculation of the index of a vector field Lemma The definition of I i does not depend on the curve C. Lemma The definition of I i does not depend on the choice of frame {e 1, e 2 }. More precisely, let S r = B r be the boundary of a disk of radius r and center p, and consider the frame {ē 1, ē 2 } of the definition. Then, the limit 1 lim r 0 ω 12 = 2π Īi S r exists, and Īi = I i. Lemma The definition of I i does not depend on the metric. We will postpone the agreement between this definition and the definition by triangulation until the last section. Example Let M be a subset of the xy-plane in R 3. In Figure 4, we take the vector field such as the singularity type of the vector field is sink. Since the definition is independent of the frame choice, we can set the simplest frame for this example.by taking e 1 as a fixed vector field (in the Figure 4, we take the vector field pointing to the right), we can see the value of the angle function on several points. For example, at p the angle is π and at q the angle is 3 2π. It can be easily seen that the angle function increasing by 2π. Hence, the index of the vector field at the singularity is 1. Similarly, if we have the vector field as Figure 5, we can see that the angle function decreasing by 2π. Then, the index of this singularity is -1. In general this fact is also true, we will come back to this in the last chapter. The fact will be used to show the agreement between the the definition by using triangulation and the definition from indices. 7

8 Figure 5: The calculation of the index of a vector field 3 Definition by Gaussian Curvature, The Gauss- Bonnet Theorem The Gaussian Curvature is an important notion in Geometry to measure how similar a local surface to hyperbola, cylinder or sphere. We start by introducing the local Gauss map. We start with a lemma from Elie Cartan. Lemma 3.1. Let V be n-dimensional vector space and let {ω i }, i = 1,..., r be linearly independent linear forms in V. If there exists forms {θ i }, i = 1,..., r such that r ω i θ i = 0 then r θ i = a ij ω j, with a ij = a ji Since ω 3 = 0, then 0 = d(ω 3 ) = ω 1 ω 13 + ω 2 ω 23. It follows from the Cartan s lemma that ω 13 = h 11 ω 1 + h 12 ω 2 ω 23 = h 21 ω 1 + h 22 ω 2, where h 12 = h 21. We name the map e 3 : U S 2 as the local Gauss map. As long as M is oriented, the map can be defined in the whole manifold. We know that de 3 = ω 31 e 1 + ω 32 e 2. So, for v = a 1 e 1 + a 2 e 2 T p M, the map can be written as ( ) ( ) h11 h de 3 (v) = 12 a1. h 21 h 22 a 2 Since the matrix is symmetric, then it has two eigenvalues λ 1, λ 2 and orthogonal eigenvectors. 8

9 Definition 3.2. We define the Gaussian curvature to be K = det(de 3 ) p = λ 1 λ 2 = h 11 h 22 h Remark 3.3. One important part of the Gauss-Bonnet theorem is the fact that dω 12 = ω 13 ω 32 = (h 11 h 22 h 2 12)ω 1 ω 2. This fact is also used to prove the Gauss theorem that states that K only depends on the induced metric of M, that is if two different immersions induce the same metric then both Gaussian curvatures from these immersions are the same. We will not prove the theorem in this work. For the boundary-less surface, the Gauss-Bonnet theorem takes the following form. Theorem 3.4. Let M be a two-dimensional compact oriented differentiable manifold. Let X be a differentiable vector fields with isolated singularities. Then, for any Riemannian metric on M, Kσ = 2πχ X (M). M where K is the Gaussian curvature of the metric and σ is its element of area. Remark 3.5. Note that since dω 12 = dω 12 we have Kω 1 ω 2 = K ω 1 ω 2. We also stated in the proof of the above theorem that ω 1 ω 2 = ω 1 ω 2. It follows that K = K or that K does not depend on the moving frame. Note that Kσ does not depend on the vector field X, therefore we can drop the subscript X from the Euler characteristic. Another surprising fact that is that lemma 2.12 gives that χ(m) does not depend on the metric, which gives us that Kσ does not depend on the Riemannian metric. M Then, this theorem gives us another alternative definition of Euler Characteristic which agree with the definition by indices. The alternative definition is χ(m) = 1 Kσ. 2π Example 3.6. By using this definition we can prove the fact that there exists no Riemannian metric on a torus T such that the Gaussian Curvature K is nonzero and does not change sign on T. We know from the example in section 1 that the Euler Characteristic from triangulation of T is 0. In the last chapter we will see that the Euler Characteristic from indices is the same as of one from the triangulation. Hence, for any Riemannian metric we have 0 = χ(t ) = 1 2π T Kσ which concludes our proof. We know that a torus T is a product of two circles. By choosing a vector field with unit length tangent to the direction of one of the circle (Figure 6), we know that if a manifold is a torus then there exists a nowhere zero differentiable vector field. Therefore, if M is homeomorphic to a torus, then there exists a nowhere zero differentiable vector field. The converse M 9

10 Figure 6: The nowhere zero vector field on a torus is also true. Let M be a a manifold such that there exists a nowhere zero differentiable vector field on it. By the definition of Euler Characteristic for indices, we know that the number is equal to 0. By using the fact that if two manifolds have the same Euler Characteristic then they are homeomorphic, we have that M is homeomorphic to a torus. 4 Definition from Morse Theorem Here, similar to the definition from indices, we will also use the concept of vector field. Let X be a differentiable vector field on M and g is a local parametrization of p = g(0, 0). Let Y be a differentiable vector field on R 2 such that g(y ) = X. Then, Y = α(x, y) x + β(x, y) y. A singularity point of X is called simple if ( ( α x ) 0 A g = ( β x ) 0 ( α y ) 0 ( β y ) 0 ) 0. Recall the example given in section 1. The following proposition can be considered a generalization. Proposition 4.1. Let p M be a singular point of a differentiable vector field X on M. Then the index of X at p is either +1 (if the determinant of the linear part of X is positive) or -1 (if the determinant of the linear part of X is negative). We can use this proposition to show the equivalence of the definitions of Euler Characteristic from triangulation and from indices. Let M be a compact orientable differentiable manifold, and consider its triangulation. Define a differentiable vector field with singularity points at each vertex of the triangles, at the middle of each edge, and in each center of the triangles. Let the vector field be indicated by its trajectories as in the Figure 7. Then, we know from the above proposition that the index of the singularity at each vertex and center of the triangles is 1. Also from the above proposition, we know that the 10

11 Figure 7: An example of triangulation (left), The vector field s trajectories for such triangulation (right) singularity at the middle of each edge contribute -1 to the Euler Characteristic. Therefore, the first two definitions of Euler Characteristic agree. From Gauss-Bonnet theorem we can conclude that the definition of Euler Characteristic does not depend on the triangulation. This result is commonly known as the Poincaré-Hopf theorem. We will now present the fourth alternative definition of Euler Characteristic. Let f : M R be differentiable. We define the vector field grad f by gradf(p), v = df p (v), for any v T p M. The proposition below, associate the concept of critical points of a function with the singularities of its gradient vector field. Proposition 4.2. Let p M be a critical point of a differentiable function f : M R where M is Riemannian. Then p is nondegenerate critical point of of f if and only if p is a simple singularity of grad f. By two propositions above and The Gauss-Bonnet theorem, we have our fourth definition of Euler Characteristic. If all critical points of f are nondegenerate, then the Euler Characteristic is equal to χ(m) = M s + m, where M, m and s are the number of points of maximum, minimum and saddle, respectively, of f. Example 4.3. Let x : M R 3 is an immersion of a two-dimensional differentiable manifold M, and let h ν (p) = p, ν for each p M. Then h ν measures the height of x(p) relative to a plane through the origin an perpendicular to ν. Then dh ν = 0 at p if and only if dh ν (w) = 0 for all w T p M at p. Since dx(p)(w) T p M, then if T p M ν (in R 3 ) then dh ν (w) = 0 for all w T p M at p which means that p is a critical point of h ν. Conversely, if p is a critical point of h ν then dx(p)(w) ν for all w T p M (in R 3 ). Using this fact, we can compute the Euler Characteristic of a torus with n holes. By choosing ν as in Figure 8, we see that p0 is a minimum point, pn + 1 is a maximum point and p0, p0, p1, p1,..., pn, pn are saddle points. Therefore, it follows from the Morse theorem, the Euler Characteristic of the torus with n holes is 2 2n. 11

12 Figure 8: The height function on a torus with n holes (seen on a projection to a plane) References [1] P. do Carmo, M., [1976]Differential Geometry of Curves and Surfaces, Prentice Hall. [2] P. do Carmo, M., [19794] Differential Forms and Applications, Springer. [3] Montiel, S., Ros, A., [2005]Curves And Surfaces, AMS Bookstore. [4] Ueno, K., Shiga, K., Morita,S.,Tyler, E., [2003] A mathematical gift: The Interplay Between Topology, Functions, Geometry, and Algebra, AMS Bookstore. 12