104 V Morse FunctionsV.1 Generic Smooth FunctionsMany, perhaps most questions in the sciences and engineering can be posedin terms of real-valued functions. General such functions are a nightmareand continuous functions are not much better. Even smooth functions canbe exceedingly complicated but when they are restricted to being generic theybecome intelligible.The upright torus. We start with an example that foreshadows many of theresults on generic smooth functions in an intuitive manner. Let M be the two-dimensional torus and f(x) the height of the point x ∈ M above a horizontalplane on which the torus rests, as in Figure V.1. We call f : M → R a heightuvwzFigure V.1: The vertical height function on the torus with critical points u, v, w, zand level sets between their height values.function. Each real number a has a preimage, f−1(a), which we refer to as alevel set. It consists of all points x ∈ M with height a. Accordingly, the sublevelset consists of all points with height at most a,Ma= f−1(−∞, a] = {x ∈ M | f(x) ≤ a}.We are interested in the evolution of the sublevel set as we increase the thresh-old. Critical events occur when a passes the height values of the points u, v, w, zin Figure V.1. For a < f(u) the sublevel set is empty. For f (u) < a < f(v) itis a disk, which has the homotopy type of a point. For f (v) < a < f(w) theV.1 Generic Smooth Functions 105sublevel set is a cylinder. It has the homotopy type of a circle which we imag-ine is obtained by gluing the two ends of an interval to the disk which is thenshrunk to a point. For f (w) < a < f(z) the sublevel set is a capped torus. Ithas the homotopy type of a figure-8 obtained by gluing the two ends of anotherinterval to the cylinder which is then shrunk to a circle. Finally, for f (z) < awe have the complete torus. It is obtained by gluing a disk to the capped torus.Figure V.2 illustrates the three intermediate stages of the evolution. We needbackground in differential topology to explain in what sense this evolution ofthe sublevel set is representative of the general situation.Figure V.2: Going from a disk to a cylinder is homotopically the same as attachinga 1-handle. Similarly, going from the cylinder to the capped torus is homotopicallythe same as attaching another 1-handle.Smooth functions. Let M be a smooth d-manifold, that is, M has an atlasof coordinate charts each diffeomorphic to an open ball in Rd. We recall that adiffeomorphism is a homeomorphism that is smooth in both directions. Denotethe tangent space at a point x ∈ M by TMx. It is the d-dimensional vectorspace consisting of all tangent vectors of M at x. A smooth mapping to anothersmooth manifold, f : M → N, induces a linear map between the tangentspaces, the derivative Dfx: TMx→ TNf (x). We are primarily interestedin real-valued functions for which N = R. Accordingly, we have linear mapsDfx: TM → TRf (x). The tangent space at a point of the real line is again areal line, so this is just a fancy way of saying that the derivatives are real-valuedlinear maps on the tangent spaces. Being linear, the image of such a map iseither the entire line or just zero. We call x ∈ M a regular point of f if Dfxis surjective and we call x a critical point of f if Dfxis the zero map. If wehave a local coordinate system (x1, x2, . . . , xd) in a neighborhood of x then xis critical iff all its partial derivatives vanish,∂f∂x1(x) =∂f∂x2(x) = . . . =∂f∂xd(x) = 0.The image of a critical point, f(x), is called a critical value of f . We use secondderivatives to further distinguish between different types of critical points. The106 V Morse FunctionsHessian of f at the point x is the matrix of second derivatives,H(x) =∂2f∂x1∂x1(x)∂2f∂x1∂x2(x) . . .∂2f∂x1∂xd(x)∂2f∂x2∂x1(x)∂2f∂x2∂x2(x) . . .∂2f∂x2∂xd(x).........∂2f∂xd∂x1(x)∂2f∂xd∂x2(x) . . .∂2f∂xd∂xd(x).A critical point x is non-degenerate if the Hessian is non-singular, that is,det H(x) 6= 0. The points u, v, w, z in Figure V.1 are examples of non-degenerate critical points. Examples of degenerate critical points are x1= 0of the function f : R → R defined by f (x1) = x31and (x1, x2) = (0, 0) off : R2→ R defined by f(x1, x2) = x31− 3x1x22. The degenerate critical point inthat latter example is often referred to as a monkey saddle. Indeed, the graphof the function in a neighborhood goes up and down three times, providingconvenient rest for the two legs as well as the tail of the monkey.Morse functions. At a critical point all partial derivatives vanish. A lo-cal Taylor expansion has therefore no linear terms. If the critical point isnon-degenerate then the behavior of the function in a small neighborhood isdominated by the quadratic terms. Even more, we can find local coordinatessuch that there are no higher-order terms.Morse Lemma. Let u be a non-degenerate critical point of f : M → R.There are local coordinates with u = (0, 0, . . . , 0) such thatf(x) = f (u) − x21− . . . − x2p+ x2p+1+ . . . + x2dfor every point x = (x1, x2, . . . , xd) in a small neighborhood of u.The number of minus signs in the quadratic polynomial is the index of thecritical point, index(u) = p. The index classifies the non-degenerate criticalpoints into d + 1 basic types. For a 2-manifold we have three types, minimawith index 0, saddles with index 1, and maxima with index 2. Examples ofall three types can be seen in Figure V.1. In Figure V.3 we display them byshowing the local evolution of the sublevel set. A consequence of the MorseLemma is that non-degenerate critical points are isolated. This implies that aMorse function on a compact manifold has at most a finite number of criticalpoints. To contrast this with a function that is not Morse take the heightfunction of a torus, similar to Figure V.1 but placing the torus sideways, theway it would naturally rest under the influence of gravity. This height functionV.1 Generic Smooth Functions 107Figure V.3: From left to right: the local pictures of a minimum, a saddle, a maximum.Imagine looking from above with the shading getting darker as the function shrinksaway from the viewpoint.has an entire circle of minima and another circle of maxima. All these criticalpoints are degenerate and their index is not defined.Definition. A smooth function on a manifold, f : M → R, is a Morsefunction if (i) all critical points are non-degenerate, and (ii) the critical
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