Arbitrary Coefficients Topology for Computing

5.1 Tangent Spaces

In this chapter, we will generally assume that M is a smooth, compact, 2- manifold without boundary, or a surface. We will also assume, for simplicity, that the manifold is embedded in R 3 , that is, M ⊂ R 3 without self-intersections. The embedded manifold derives subspace topology and a metric from R 3 . These assumptions are not necessary, however. The ideas presented in this chapter generalize to higher dimensional abstract manifolds with Riemannian metrics. We begin by attaching tangent spaces to each point of a manifold. As al- ways, we derive our notions about manifolds from the Euclidean spaces. Definition 5.1 T p R 3 A tangent vector v p to R 3 consists of two points of R 3 : its vector part v and its point of application p. The set T p R 3 consists of all tangent vectors to R 3 at p and is called the tangent space of R 3 at p . Note that R 3 has a different tangent space at every point. Each tangent space is a vector space isomorphic to R 3 itself. We may also attach a vector space to each point of a manifold. Definition 5.2 T p M Let p be a point on M in R 3 . A tangent vector v p to R 3 at p is tangent to M at p if v is the velocity of some curve in M. The set of all tangent vectors to M at p is called the tangent plane of M at p and is denoted by T p M. Recall from Chapter 2 that a 2-manifold is covered with a number of charts, which map the neighborhood of a point to an open subset of R 2 . Each map is a homeomorphism, and we may parameterize the manifold using the inverses of these maps, which are often called patches. Theorem 5.1 Let p ∈ M ⊂ R 3 , and let ϕ be a path in M such that ϕu , v = p. A tangent vector v to R 3 at p is tangent to M iff v can be written as a linear combination of ϕ u u , v and ϕ v u , v . In other words, the tangent plane at a point of the manifold is a two-dimensional vector subspace of the tangent space T p R 3 , as shown in Figure 5.1. Based on the properties of derivatives, the tangent plane T p M is the best linear approx- imation of the surface M near p. Given tangent planes, we may select vectors at each point of the manifold to create a vector field. Definition 5.3 vector field A vector field or flow on V is a function that as- signs a vector v p ∈ T p M to each point p ∈ M. p T M M p v Fig. 5.1. The tangent plane T p M to M at p with tangent vector v ∈ T p M.

5.2 Derivatives and Morse Functions

Intuitively, a tangent vector gives us a direction to move on a surface. If we have a real-valued smooth function h defined on a manifold, we may ask how h changes as we move in the direction specified by the tangent vector. Definition 5.4 derivative Let v p ∈ T p M and let h : M → R. The derivative v p [h] of h with respect to v p is the common value of ddth ◦ γ0, for all curves γ ∈ M with initial velocity v p . Here, we are using the Euclidean metric to measure the length of v p . This definition is a generalization of the derivative of functions on R, except that now we can travel in many different directions for different rates of changes. The differential of a function captures all rates of change of h in all possible directions on a surface. The possible directions are precisely vectors in T p M. Definition 5.5 differential The differential dh p of h : M → R at p ∈ M is a linear function on T p M such that dh p v p = v p [h], for all tangent vectors v p ∈ T p M. We may view the differential as a machine that converts vector fields into real- valued functions O’Neill, 1997. Given a function h and a surface M, we are interested in understanding the geometry h gives our manifold. We travel in all directions, starting from a point p, and note the rate of change. If there is no change in any direction, we have a found a special point, critical to our understanding of the geometry. Definition 5.6 critical A point p ∈ M is critical for map h : M → R if dh p is the zero map. Otherwise, p is regular. To further classify a critical point, we have to look at how the function’s deriva- tive changes in each direction. The Hessian is a symmetric bilinear form on the tangent space T p M, measuring this change. Like the derivative, it is in- dependent of the parameterization of the surface. We may state it explicitly, however, given local coordinates on the manifold. Definition 5.7 Hessian Let x , y be a patch on M at p. The Hessian of h : M → R is H p = ∂ 2 h ∂x 2 p ∂ 2 h ∂y∂x p ∂ 2 h ∂x∂y p ∂ 2 h ∂y 2 p . 5.1 The definition gives the Hessian in terms of the basis ∂ ∂x p, ∂ ∂y p for T p M. We may classify the critical points of a manifold, and an associated real-valued function, using the Hessian. Definition 5.8 degeneracy A critical point p ∈ M is nondegenerate if the Hessian is nonsingular at p, i.e., det H p = 0. Otherwise, it is degenerate. We are interested in functions that only give us nondegenerate critical points. Definition 5.9 Morse function A smooth map h : M → R is a Morse func- tion if all its critical points are nondegenerate. Any twice differentiable function h may be unfolded to a Morse function. That is, there is Morse a function that is as close to h as we would like it to be. Sometimes, the definition of Morse functions also requires that the critical values of h, that is—values h takes at its critical points—are distinct. We do not need this requirement here.

5.3 Critical Points

We may, in fact, fully classify the critical points of a Morse function by the geometry of their neighborhood. We do so for a 2-manifold in this section. Lemma 5.1 Morse lemma It is possible to choose local coordinates x , y at a critical point p ∈ M so that a Morse function h takes the form: h x, y = ±x 2 ± y 2 . 5.2 Figure 5.2 shows the four possible graphs of h, near the critical point 0, 0. The existence of these neighborhoods means that the critical points are iso- lated : They have neighborhoods that are free of critical points. Using the Morse characterization, we name the critical points using an index. a x 2 + y 2 b −x 2 + y 2 c x 2 − y 2 d −x 2 − y 2 Fig. 5.2. The neighborhood of a critical point 0, 0 of index 0, 1, 1, and 2, from the left, corresponding to the possible forms of h. a is a minimum, b and c are saddles, and d is a maximum. Definition 5.10 index The index ip of h at critical point p ∈ M is the num- ber of minuses in Equation 5.2. Equivalently, the index at p is the number of the negative eigenvalues of H p. Definition 5.11 minimum, saddle, maximum A critical point of index 0, 1, or 2, is called a minimum, saddle, or maximum, respectively. The Morse lemma states that the neighborhood of a critical point of a Morse function cannot be more complicated than those in Figure 5.2. For example, the neighborhood shown in Figure 5.3 is not possible. A point with this neigh- borhood is often called a monkey saddle, as its geometry as a saddle allows for a monkey’s tail. Fig. 5.3. The monkey saddle at 0, 0 is a degenerate critical point.

5.4 Stable and Unstable Manifolds

The critical points of a Morse function are locations on a 2-manifold where the function is stationary. To fully understand a Morse function, we need to extract more structure. To do so, we first define a vector field called the gradient. Definition 5.12 gradient Let γ be any curve passing through p, tangent to v p ∈ T p M. The gradient ∇h of a Morse function h is d γ dt · ∇h = d h ◦ γ dt . 5.3 In the general setting, the inner product above is replaced by an arbitrary Riemannian metric Boothby, 1986. The gradient is related naturally to the derivative, as v p [h] = v p · ∇hp. It is always possible to choose coordinates x, y so that the tangent vectors ∂ ∂x p, ∂ ∂y p are orthonormal with respect to the chosen metric. For such coordinates, the gradient is given by the familiar formula ∇h = ∂h ∂x p, ∂h ∂y p. The gradient of a Morse function h is a vector field on M. We integrate this vector field, in order to decompose M into regions of uniform flow. Definition 5.13 integral line An integral line γ : R → M is a maximal path whose tangent vectors agree with the gradient, that is, ∂ ∂s p s = ∇hps for all s ∈ R. We call org p = lim s →−∞ p s the origin and dest p = lim s → + ∞ p s the destination of the path p. Each integral line is open at both ends, and the limits at each end exist, as M is compact. Note that a critical point is an integral line by itself. Theorem 5.2 Integral lines have the following properties: a Two integral lines are either disjoint or the same. b The integral lines cover all of M. c And the limits org p and dest p are critical points of h. The properties follow from standard differential calculus. Definition 5.14 stable and unstable manifolds The stable manifold S p and the unstable manifold U p of a critical point p are defined as S p = {p} ∪ {y ∈ M | y ∈ im γ, dest γ = p}, 5.4 U p = {p} ∪ {y ∈ M | y ∈ im γ, org γ = p}, 5.5 where im γ is the image of the path γ ∈ M. Both sets of manifolds decompose M into open cells. Definition 5.15 open cell An open d-cell σ is a space homeomorphic to R d . We can predict the dimension of the open cell associated to a critical point p. Theorem 5.3 The stable manifold S p of a critical point p with index i = ip is an open cell of dimension dim S p = i. The unstable manifolds of h are the stable manifolds of −h as ∇−h = −∇h. Therefore, the two types of manifolds have the same structural properties. That is, the unstable manifolds of h are also open cells, but with dimension dimU p = 2 − i, where i is the index of a critical point. The closure of a stable or unstable manifold, however, is not necessarily homeomorphic to a closed ball. We see this in Figure 5.4, where a stable 2-cell is pinched at a minimum. By the properties in Theorem 5.2, the stable manifolds are pairwise disjoint and decompose M into open cells. The cells form a complex, as the bound- ary of every cell S a is a union of lower dimensional cells. We may view a cellular complex as a generalization of a simplicial complex, where we allow for arbitrarily shaped cells and relax restrictions on how they are connected to each other. The unstable manifolds similarly decompose M into a complex dual to the complex of stable manifolds: For a , b ∈ M, dim Sa = 2 − dimUa and Sa is a face of S b iff Ub is a face of Ua.