Theory

Gradient, critical points; Hessian, non-degenerate points

Suppose we are given a smooth function $f:{\mathbb{R}}^d\to \mathbb{R}$.

The differential matrix of $f$ in this case is (the transpose of) the gradient of $f$:

$$\nabla f(p)=[d_{p}f{]}^{\mathsf{T}}={\left(\begin{array}{c}\frac{\partial f}{\partial x_1}(p),\frac{\partial f}{\partial x_2}(p),\dots ,\frac{\partial f}{\partial x_d}(p)\end{array}\right)}^{\mathsf{T}}$$

By the definition of regular point, a point $p\in {\mathbb{R}}^d$ is regular for $f$ when the gradient vector is not $\vec{0}$, i.e. when $\frac{\partial f}{\partial x_i}(p)\ne 0$ for some coordinate $x_i$. The point $p$ is called critical when it is not regular.

(Note: $f$ is smooth by hypothesis, so the components of $\nabla f$ are everywhere defined, and so critical points are points where $\nabla f=\vec{0}$.)

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The differential $[d_{p}f]$ of a smooth function $f:{\mathbb{R}}^d\to \mathbb{R}$ is itself a smooth function ${\mathbb{R}}^d\to {\mathbb{R}}^d$. We may thus take the differential of the differential, and we obtain the Hessian matrix of second partial derivatives:

$$[H_p(f)]=\left(\begin{array}{cccc}\frac{{\partial }^{2}f}{\partial x_1^2}& \frac{{\partial }^{2}f}{\partial x_1\partial x_2}& \cdots & \frac{{\partial }^{2}f}{\partial x_1\partial x_n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{{\partial }^{2}f}{\partial x_n\partial x_1}& \frac{{\partial }^{2}f}{\partial x_n\partial x_2}& \cdots & \frac{{\partial }^{2}f}{\partial x_n^2}\end{array}\right)$$

The Hessian matrix is always a symmetric square matrix because $\frac{{\partial }^{2}f}{\partial x_i\partial x_j}=\frac{{\partial }^{2}f}{\partial x_j\partial x_i}$. This implies that all eigenvalues are real numbers (a case of the Spectral Theorem).

Non-degenerate critical point

A critical point $p$ for a function $f:{\mathbb{R}}^d\to \mathbb{R}$ is non-degenerate when $[H_p(f)]$ is a non-singular matrix.

Equivalently, when:

• $\det [H_p(f)]\ne 0$
• All eigenvalues of $[H_{p}f]$ are nonzero

Index of a critical point

Given a non-degenerate critical point $p$, the index of $p$ is the number of negative eigenvalues of $[H_p(f)]$.

Isolated critical point

A critical point $p$ for a function $f:{\mathbb{R}}^d\to \mathbb{R}$ is isolated when an open neighborhood of $p\in {\mathbb{R}}^d$ can be found such that $p$ is the only critical point of $f$ in this neighborhood.

Morse functions on manifolds

Morse theory is about the use of a function $f:M\to \mathbb{R}$ to study the topology of the manifold $M$ in terms of the critical points of $f$. One sets up a function $f$ that has nondegenerate critical points with distinct critical values, called a Morse function. To make sense of this setup we need first to define smooth functions $f:M\to \mathbb{R}$.

Smooth function on a manifold

A function $f:M\to \mathbb{R}$ on a smooth manifold $M\subset {\mathbb{R}}^d$ is smooth if it is the restriction of a smooth function $\tilde{f}:U\to \mathbb{R}$ where $U$ is an open set in ${\mathbb{R}}^d$ with $M\subset U\subset {\mathbb{R}}^d$.

This means: $f=\tilde{f}{|}_M$ and $\tilde{f}$ has continuous partial derivatives of all orders.

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Recall local coordinates:
For any $p\in M$ there is a set $V\ni p$ open in ${\mathbb{R}}^n$ and $U$ open in $M$ with

$$\phi :V\to U$$

a parametrization (diffeomorphism to its image). The ${\mathbb{R}}^n$-coordinates of images of the inverse ${\phi }^{-1}$ give local coordinates on $U$.

Critical and non-degenerate, from a manifold

Let $f:M\to \mathbb{R}$ be a smooth function on a smooth $n$-manifold in ${\mathbb{R}}^d$, and $\phi :V\to U$ a parametrization of the open neighborhood $U$ of $p$. So $f\circ \phi :V\to \mathbb{R}$ for $V\subset {\mathbb{R}}^n$.

Choose any $q\in U$. Set $x={\phi }^{-1}(q)$.

We say $q$ or $x$ is regular when $\nabla (f\circ \phi )(x)\ne \vec{0}$ and critical when $\nabla (f\circ \phi )(x)=\vec{0}$.

We say $q$ or $x$ is non-degenerate when $[H_x(f\circ \phi )]$ is a non-singular matrix (and degenerate otherwise).

Notice that this definition of regularity appears to depend on the parametrization $\phi$. But in fact, the definition is independent of the choice of $\phi$:

Proposition - Regularity is independent of parametrization

Suppose we are given two parametrizations ${\phi }_1$ and ${\phi }_2$ of a coordinate patch $U\subset M$.

Take an arbitrary $q\in U$ and assume it is regular according to ${\phi }_1$.

We show it must be regular according to ${\phi }_2$.

This will complete the proof because if it is critical according to ${\phi }_1$, and not critical according to ${\phi }_2$, then it is regular according to ${\phi }_2$, and the same argument can be run after switching ${\phi }_1\leftrightarrow {\phi }_2$.

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Define the ‘transition map’ $\psi ={\phi }_1^{-1}\circ {\phi }_2$.

This $\psi$ is a map $V_1\to V_2$ where ${\phi }_1:V_1\to U$ and ${\phi }_2:V_2\to U$ for $V_i\subset {\mathbb{R}}^n$ open balls.

Notice that $(f\circ {\phi }_2)=(f\circ {\phi }_1)\circ \psi$.

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Apply chain rule:

$$\nabla (f\circ {\phi }_2)=\nabla (f\circ {\phi }_1\circ \psi )=\nabla (f\circ {\phi }_1)\circ [d\psi ]$$

Here $[d\psi ]$ is the Jacobi matrix of $\psi$.

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Argue that $[d\psi ]$ is invertible:

We have that $[d{\phi }_1]$ and $[d{\phi }_2]$ are both isomorphisms from ${\mathbb{R}}^n$ to the tangent space of $M$ at $q$.

Then $[d\psi ]$ is the composition of $[d{\phi }_2]$ with the inverse of $[d{\phi }_1]$ restricted to the tangent space of $M$ at $q$. (Prior to restriction $[d{\phi }_1]$ is not even a square matrix.)

Therefore $[d\psi ]$ is invertible.

Therefore, at any point:

$$\nabla (f\circ {\phi }_2)=0⟺\nabla (f\circ {\phi }_1\circ \psi )=0$$

So a point is regular for ${\phi }_1$ if and only if it is regular for ${\phi }_2$.

Why define regularity using parametrizations?

Since the choice of parametrization has no effect on the classification of points as regular or critical, one may wonder why the parametrization is needed at all.

But the concept of regularity does involve the manifold’s embedding, and not just the values of $f$ in the ambient space around the manifold.

A point $q\in M$ is regular for $f$ when there is some first-order infinitesimal deviation of $q$ within $M$ which causes a first-order deviation in $f$. When $q$ is critical, the induced change in $f$ is second order at most.

For example, given a smooth function $f:{\mathbb{R}}^n\to \mathbb{R}$, a level surface $M$ is (frequently) an $(n-1)$-manifold every point of which is regular for $f:{\mathbb{R}}^n\to \mathbb{R}$, but no point of which is regular for $f{|}_M:M\to \mathbb{R}$.

Tangent space $T_{p}M$

Given a manifold $M\subset {\mathbb{R}}^d$, we define the tangent space to $M$ at $p\in M$, written $T_{p}M$, as the image of $[d_p\phi ]$ in ${\mathbb{R}}^d$ for $\phi$ any parametrization of a neighborhood of $p$.

Comparing gradient functions

Notice:

• $\nabla \tilde{f}$ is a map from ${\mathbb{R}}^d$, and has $d$ coordinates
• $\nabla (f\circ \phi )$ is a map from ${\mathbb{R}}^n$, and has $n$ coordinates, one for each dimension of $M$

In general: ${\nabla }_p(f\circ \phi )$ is the restriction of ${\nabla }_p\tilde{f}$ to $T_{p}M$ using columns of $[d\phi ]$ as a basis.

(Supposing $M\subset {\mathbb{R}}^d$ and $\tilde{f}$ an extension of $f:M\to \mathbb{R}$ to $U$ with $M\subset U\subset {\mathbb{R}}^d$.)

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Morse Lemma

Suppose $q$ is a non-degenerate critical point of index $i$ of a smooth function $f:M\to \mathbb{R}$.

Then:

$$f\circ \phi =x_1^2+\cdots +x_{n-i}^2-x_{n-i+1}-\cdots -x_n^2$$

for some parametrization $\phi$ whose local coordinates are $x_1,\dots ,x_n$.

Morse functions

A Morse function $f:M\to \mathbb{R}$ is a smooth function from a manifold $M\subset {\mathbb{R}}^d$ to $\mathbb{R}$ which satisfies:

1. All critical points of $f$ are non-degenerate.
2. The values of $f$ at its critical points are all distinct.

Morse functions are dense

Morse functions are dense in the space of smooth functions.

There is a small perturbation of any smooth function that is a Morse function.

Morse functions and topology

Given a Morse function $f:M\to \mathbb{R}$ for a manifold $M$, define:

• $M_C=f^{-1}(C)$ the level set at $C$
• $M_{\le C}=f^{-1}((-\infty ,C])$ the sublevel set under $C$

By continuously increasing the value of $C$ (starting at $-\infty$), we find that the topology of $M_{\le C}$ changes exactly when $C$ passes a critical point of $f$.

Sublevel set builds topology

Up to homotopy equivalence, as $C$ passes through a critical point of index $k$, the sublevel set changes by the attachment of a $k$-ball along its boundary.

For example:

• Pass a point of index $2$ $⇝$ glue on a disk, a lid opening downwards
• Pass a point of index $1$ $⇝$ glue on an arc (up to homotopy)
• Pass a point of index $0$ $⇝$ create a new disk, a basket opening upwards