Morse Functions

Abstract

We define Morse functions and prove the basic results (existence and genericity, Morse lemma …)

Exercises

Exercise 1

Let U be an open subset of R ⁿ and let $f:U\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbf{R}$ be a function. Let V be an open subset of R ⁿ and let $\varphi:V\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}U$ be a diffeomorphism. Compute (d ²(f∘φ)) _y (for y∈V).

Let M be a manifold and let $g:M\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbf{R}$ be a function. Show that the bilinear form (d ² g) _x is well defined on the vector subspace

$$\operatorname{Ker}(dg)_x\subset T_xM. $$

Exercise 2

The cotangent vector bundle T ^⋆ V of the manifold V is endowed with its natural manifold structure (recalled in Appendix A). Verify that we can see the differential df of a function $f:V\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbf{R}$ as a section

$$df:V\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}T^\star V $$

of the cotangent vector bundle (which is an embedding of V in T ^⋆ V).

What are the critical points of f in this terminology? Show that the point a is a nondegenerate critical point of f if and only if the submanifold df(V) is transverse to the zero section at the point in question.

Prove that a nondegenerate critical point of a function is isolated (without using the Morse lemma!).

Exercise 3

(Monkey Saddle)

We consider the function

$$\begin{aligned} f:\mathbf{R}^2&\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbf{R}\\ (x,y)&\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}x^3-3xy^2. \end{aligned}$$

Study the critical point (0,0). Is it nondegenerate? Draw a few regular level sets of the function f as well as its graph.

Exercise 4

If $f:V\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbf{R}$ and $g:W\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbf{R}$ are Morse functions, then $f+g:V\times W\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbf{R}$ is also a Morse function whose critical points are the pairs of critical points of f and g.

Exercise 5

(On the Complex Projective Space)

By passing to the quotient, the function on C ⁿ⁺¹−{0} defined by

$$f(z_0,\dots ,z_{n})=\frac{\sum_{j=0}^nj\left \vert z_{j}\right \vert ^2}{\sum_{j=0}^n\left \vert z_{j}\right \vert ^2} $$

induces a function on the complex projective space P ⁿ(C), which we still denote by f.

Verify that this is a Morse function whose critical points are the points [1,0,…,0] (of index 0), [0,1,0,…,0] (of index 2) and so on up to [0,…,0,1] (of index 2n).

Readers eager to know where this function may well have come from (from [54], of course, but before that?) can turn to Remark 2.1.12.

Exercise 6

(On the sphere and on the real projective plane)

With essentially the same formula, let us now consider the function

$$\begin{aligned} f:S^2&\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbf{R}\\ (x,y,z)&\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}y^2+2z^2 \end{aligned}$$

and the resulting function $g:\mathbf{P}^{2}(\mathbf{R})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbf{R}$ on the real projective plane that follows from it by passing to the quotient. Verify that f (and therefore g) is a Morse function and show that it has six critical points (and therefore that g has three):

• Two points of index 0, the points (±1,0,0), at level 0

• Two points of index 1, the points (0,±1,0), at level 1

• Two points of index 2, the points (0,0,±1), at level 2.

Figure 1.7 shows a few level sets of this function on the sphere. The critical level set containing the two points of index 1 consists of the two circles defined by intersecting the sphere with the planes z=±x.

Fig. 1.7

A Morse function on the real projective plane

Exercise 7

In a square, draw the level sets of the height function on the “inner tube” torus (the answer is somewhere in this book).

Exercise 8

(A projective quadric)

We consider the “quadric” Q in P ³(C) defined by the equation

$$z_0^2+z_1^2+z_2^2+z_{3}^2=0.$$

Show that Q is a compact submanifold of the manifold P ³(C) of (real) dimension 4.

Let us write z _j =x _j +iy _j (for 0≤j≤3) and z=x+iy for z∈C ⁴, with x,y∈R ⁴. We fix two real numbers λ and μ and, for z∈C ⁴, we set

$$\widetilde {f}(z)=\lambda(x_0y_1-x_1y_0)+\mu(x_2y_{3}-x_{3}y_2).$$

Verify that for $\left \vert u\right \vert =1$, we have

$$\widetilde {f}(uz)=\widetilde {f}(z)$$

and deduce from it that $\widetilde {f}$ defines a function

$$f:\mathbf{P}^{3}(\mathbf{C})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbf{R}.$$

Suppose that the real numbers λ and μ occurring in the definition of f satisfy 0<λ<μ. Let g be the restriction of f to Q. Show that g is a Morse function that has a local minimum, a local maximum and two critical points of index 2.

Exercise 9

In this exercise, we propose a different proof of the existence of numerous Morse functions on a manifold, and more precisely of Theorem 1.2.5:

• (1) Using, for example, Exercise 2, show that the set of Morse functions on the compact manifold V is open for the topology, and therefore also for the topology.

• (2) Let a∈V and let U be the open set of a chart centered at a. We let denote the space of differentials of the functions defined on U (or of exact 1-forms) , seen as a subset of (for every x∈U, (df) _x is a linear functional on R ⁿ). Note that is not an open subset. Let α be a plateau function with support in U and value 1 on a compact neighborhood K of a. For , we let

  $$\begin{aligned} F:U\times(\mathbf{R}^n)^\star&\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbf{R}\\ (x, A)&\mathchoice{\longmapsto}{\mapsto}{\mapsto}{\mapsto}f(x)+\alpha(x)A\cdot x. \end{aligned}$$

  Compute³ (D ₁ F)_(x,A) for x∈K and show that F is a submersion

  $$K\times(\mathbf{R}^n)^\star \mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\mathbf{R}^n)^\star.$$

  Show that is locally transversal to $\left \{0\right \}$ (in the sense of Subsection A.3.c).

• (3) Show that every point a of V admits a compact neighborhood K such that the set of functions $f:V\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow} \mathbf{R}$ whose critical points in K are nondegenerate is dense in .

• (4) Deduce a proof of Theorem 1.2.5 from this.