Theory of Support Functions

The goal of this post is to work through the following theorem:

The manifold of 1-jets of functions $f: S^{n-1} \rightarrow \mathbb{R}$ (with its natural contact structure) is contact diffeomorphic to $ST^{*} \mathbb{R}^n \cong \mathbb{R}^n \times S^{n-1}$ , the manifold of co-oriented contact elements in $\mathbb{R}^n$ .

Recall: the natural contact structure of the space of 1-jets $J^1(\mathbb{R}^n, \mathbb{R})$ is given by $dy = pdx$ , where $p_i = \frac{\partial y}{\partial x_i}$ .

A point in $J^1(S^{n-1}, \mathbb{R})$ , the space of 1-jets of $f: S^{n-1} \mathbb{R}$ , is given by:

$(q, z = f(q), p = df|_q)$ , where $q \in S^{n-1}$ .

Since $df_q$ is the gradient of $f$ at q, it means that $df_q$ gives us the direction of fastest ascent. This direction lives in the tangent space of $S^{n-1}$ at $q$ . This tangent space is the space orthogonal to $q$ , i.e. $\{ q^{\perp} \}$ . Hence, $df|_q \in \mathbb{R}^n$ .

It follows from the structure given in $J^1(\mathbb{R}^n, \mathbb{R})$ that the natural contact structure in $J^1(S^{n-1}, \mathbb{R})$ is given by:

$dq = p dz$ .

What is the tautological contact structure on $ST^{*} \mathbb{R}^n$ ?

We denote a point in $ST^{*} \mathbb{R}^n \cong \mathbb{R}^n \times S^{n-1}$ to be $(Q, P)$ , where $Q \in \mathbb{R}^n$ and $P \in S^{n-1}$ .

The tautological contact structure is the field of hyperplanes in $ST^{*}$ such that the projection of each hyperplane to $\mathbb{R}^n$ gives us a co-oriented contact element; in this case, it would be $(Q, \mathcal{H}_P)$ ( $Q \in \mathbb{R}^n$ and $\mathcal{H}_P$ is a co-oriented hyperplane in $\mathbb{R}^n$ centered at $Q$ ). $\mathcal{H}_P$ corresponds uniquely to an oriented 1-form $dP$ , which identifies uniquely with a ray emanating from the origin, $P \in S^{n-1}$ . We can think of $P$ as one of the two rays in the space orthogonal to $\mathcal{H}_P$ .

Thus, we can see that each co-oriented contact element, $(Q, \mathcal{H}_P)$ corresponds to unique a point $(Q, P) \in \mathbb{R}^n \times S^{n-1} \cong ST^{*} \mathbb{R}^n$ .

Since $\mathcal{H}_P$ can be thought of as the space of vectors orthogonal to $P$ emanating from $Q$ , it follows that:

$\mathcal{H}_P = \{ dQ \in T_Q \mathbb{R}^n : dQ \cdot P = 0 \}$

The corresponding hyperplane in the tautological contact structure is the pre-image of this space in $ST^{*} \mathbb{R}^n$ under the natural projection $\pi: ST^{*} \cong \mathbb{R}^n \times S^{n-1} \rightarrow \mathbb{R}^n$ . This hyperplane is the set of vectors, $(dQ, dP)$ emanating from $(Q, P)$ such that $dQ \cdot P = 0$ , i.e.

$\pi^{-1} (\mathcal{H}_P) = \{ (dQ, dP) \in T(ST^{*} \mathbb{R}^n) : dQ \cdot P = 0 \} := \tilde{\mathcal{H}}_P$

Hence, the tautological contact structure in $ST^{*} \mathbb{R}^n$ is given by $dQ \cdot P = 0$ .

Claim:

$\Psi: J^1(S^{n-1}, \mathbb{R}) \rightarrow ST^{*} \mathbb{R}^n$ given by

$\Psi: (q, z, p) \mapsto (qz + p, q)$

is a contact diffeomorphism.

Proof:

Since $q \in S^{n-1}$ , $z \in \mathbb{R}$ , and $p \in \{ q^{\perp} \} \cong \mathbb{R}^{n-1}$ , the map is well-defined and that $\{qz + p: z \in \mathbb{R}, p \in \{ q^{\perp} \} \cong \mathbb{R}^n$ . Hence, $\Psi$ is bijective. It is easy to see that $\Psi$ is smooth. Hence, it is a diffeomorphism.

We now show that $\Psi$ maps the natural contact structure in $J^1$ into the tautological contact structure $ST^{*}$ .

It suffices to show that given $(dp, dz, dq)$ at $(p, z, q)$ , the corresponding $(dQ, dP)$ at $(Q, P)$ (i.e. the image of $d\Psi(dp, dz, dq)$ ) satisfies $P \cdot dQ = 0$ .