Examples

Same critical point, different index

Same critical point, different index

Consider these functions from ${\mathbb{R}}^2$ to $\mathbb{R}$:

$$\begin{array}{rl}f(x,y)& =x^2+y^2\\ g(x,y)& =x^2-y^2\\ h(x,y)& =-x^2-y^2\end{array}$$

The point $p=(0,0)$ is the only critical point for each of these functions.

The Hessian matrices at $p$ are:

$$[H_p(f)]=\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right)[H_p(g)]=\left(\begin{array}{cc}2& 0\\ 0& -2\end{array}\right)[H_p(h)]=\left(\begin{array}{cc}-2& 0\\ 0& -2\end{array}\right)$$

So $p$ has index $0$ for $f$, index $1$ for $g$, and index $2$ for $h$.

Height function

Height function

Suppose $\tilde{f}:{\mathbb{R}}^3\to \mathbb{R}$ by $\tilde{f}(x,y,z)=z$.

Then $\nabla \tilde{f}=(0,0,1)$. Since $\nabla (f\circ \phi )$ is the restriction of $\nabla \tilde{f}$ to $T_{p}M$, we see that $p$ is critical for $f$ precisely when $T_{p}M$ is normal to $(0,0,1)$.

In other words: critical points of $f$ are points where the tangent space to $M$ is a horizontal plane.

This reasoning generalizes to higher dimensions ${\mathbb{R}}^d$ where one dimension is chosen as ‘vertical’, and horizontal planes become hypersurfaces normal to the vertical unit vector.

Height function on a torus

Height function on a torus

Suppose $f:{\mathbb{R}}^3\to \mathbb{R}$ is the height function $f(x,y,z)=z$, and $M$ is topologically a torus, though not geometrically.

As $C$ passes through $f(P_1)$, a small disk is created in $M_{\le C}$.

Passing through $f(P_2)$ is (up to homotopy) equivalent to adding a handle to the basket below $C_2$:

Passing through $f(P_3)$ adds a new basket, and passing through $f(P_4)$ adds a handle connecting the basket:

Passing through $f(P_5)$ adds a handle creating a loop: