Notes of Alessandro Ottazzi’s lecture nr 3

1. Liouville’s theorem

Let ${f}$ be a ${C^2}$ conformal map on an open domain of ${{\mathbb R}^n}$, ${n\geq 3}$. Observe that ${f}$ is conformal if and only if at every point ${x}$, ${Df(x)}$ belongs to the conformal group ${CO(n)=\{A\in Gl(n,{\mathbb R})\,;\,\exists \mu>0}$ such that ${AA^{\top}=\mu I\}}$. In other words, ${f}$ is an automorphism of the constant ${G}$-structure, ${G=CO(n)}$.

Infinitesimal conformal transformations are of the form

$\displaystyle \begin{array}{rcl} V=\sum_{i=1}^{n}v_i \frac{\partial}{\partial x_i}, \end{array}$

where the matrix ${(\frac{\partial v_i}{\partial x_j}}$ belongs to the Lie algebra

$\displaystyle \begin{array}{rcl} \mathfrak{co}(n)=\{X\in Gl(n,{\mathbb R})\,;\,\exists \nu,\,X+X^{\top}=\nu I\}. \end{array}$

1.1. First step : prolongation and infinitesimal transformations

The prolongation is of the form ${{\mathbb R}^n \oplus \mathfrak{co}(n)\oplus\mathcal{G}^{(1)}(\mathfrak{co}(n))}$, it is isomorphic, as a Lie algebra, to ${\mathfrak{so}(1,n+1)}$. This is a classical computation. This is a Liouville theorem for vectorfields.

1.2. Second step : conformal transformations

Assume that ${f(0)=0}$. Let ${U}$ an infinitesimal conformal transformations. So is ${f_* U}$, but, a priori, ${f_* U}$ is only ${C^1}$. Mollify it. Since the equations are linear with constant coefficients, the mollified vectorfield is still conformal, thus belongs to ${\mathfrak{so}(1,n+1)}$. As a limit of such, ${f_* U \in\mathfrak{so}(1,n+1)}$. Whence a group homomorphism ${\Phi:Conf\rightarrow Aut(\mathfrak{so}(1,n+1))=O(n,1)}$.

One show that this homomorphism is injective. Indeed, assume that ${g(0)=0}$ and ${g_* U=f_* U}$ for all ${U\in \mathfrak{so}(1,n+1)}$. Then ${g^{-1}\circ f}$ maps a constant vectorfield to itself, its differential equals identity, it is a translation, therefore it equals identity.

On the Möbius group ${O(n,1)}$, ${\Phi}$ is identity, thus showing that ${f\in O(n,1)}$.

2. Tanaka prolongation theory

Let ${G\subset Gl(n,{\mathbb R})}$ be a Lie group. Assume ${G}$ leaves a vectorsubspace ${V\subset {\mathbb R}^n}$ invariant. A ${G}$-structure subsumes a plane distribution. The Singer-Sternberg prolongation works if the ${G}$-structure is integrable, which often forces the distribution to be involutive, and we do not want this.

Tanaka replaces constant ${G}$-structures by different models: left-invariant ${G}$-structures on stratified Lie groups.

Definition 1 A model space (for Tanaka) is a stratified Lie algebra

$\displaystyle \begin{array}{rcl} \mathfrak{n}=\mathfrak{g}_{-s}\oplus\cdots\oplus\mathfrak{g}_{-1}. \end{array}$

such that for all ${-s+1\leq j\leq -1}$,

$\displaystyle \begin{array}{rcl} [\mathfrak{g}_{-1},\mathfrak{g}_{j}]=\mathfrak{g}_{j-1}. \end{array}$

The corresponding Lie group ${N}$ comes equipped with a horizontal distribution ${\mathcal{H}N}$. Contact mappings (mappings that preserve ${\mathcal{H}N}$) can be viewed as automorphisms of a geometric structure.

Definition 2 An infinitesimal automorphism is a vectorfield whose local flow consists of contact transformations.

Remark 1 ${V}$ is an infinitesimal contact transformation if and only if ${[V,\Gamma(\mathcal{H}N)]\subset\Gamma(\mathcal{H}N)}$.

2.1. The full prolongation

We first define the full prolongation, i.e. Lie algebra relevant to the determination of all infinitesimal contact transformations.

Definition 3 ${Prol(\mathfrak{n})=\bigoplus_{k=-s}^{\infty}\mathfrak{g}_{k}(\mathfrak{n})}$, where ${\mathfrak{g}_{0}(\mathfrak{n})=Der_{0}(\mathfrak{n})}$ is the space of degree ${0}$ derivations, and for positive ${k}$,

$\displaystyle \begin{array}{rcl} \mathfrak{g}_{k}=\{u:\mathfrak{n}\rightarrow \mathfrak{n}\oplus\mathfrak{g}_0 (\mathfrak{n})+\cdots+\mathfrak{g}_{k-1} (\mathfrak{n})\,;\,\forall X,\,Y\in\mathfrak{n},\,u[X,Y]=u(X)(Y)-u(Y)(X)\}. \end{array}$

Note that if ${\mathfrak{g}_{k-1}(\mathfrak{n})=0}$, then ${\mathfrak{g}_{k}(\mathfrak{n})=0}$.

Definition 4 The Lie bracket structure is defined as follows. For ${u\in\mathfrak{g}_{k}(\mathfrak{n})}$, ${u'\in\mathfrak{g}_{k'}(\mathfrak{n})}$ and ${X\in\mathfrak{n}}$, set ${[u,X]=(X)}$ and inductively

$\displaystyle \begin{array}{rcl} [u,u'](X):=[u,[u',X]]-[u',[u,X]]. \end{array}$

Definition 5 Replacing ${Der_{0}(\mathfrak{n})}$ by a subalgebra ${\mathfrak{g}_0 \subset Der_0(\mathfrak{n})}$, one gets ${Prol(\mathfrak{n},\mathfrak{g}_0)}$.

2.2. Finiteness criterion

Definition 6 Define

$\displaystyle \begin{array}{rcl} \mathcal{G}^{(0)}=\{u\in\mathfrak{g}_{0}\,;\,\forall X,\,Y\in\mathfrak{n},\, u[X,Y]=0 \}. \end{array}$

Note ${\mathcal{G}^{(0)}}$ can be viewed as a subalgebra of ${\mathfrak{gl}(\mathfrak{g}_{-1})}$, and thus admits a prolongation in the sense of Singer and Sternberg.

Theorem 7 (Tanaka) ${Prol(\mathfrak{n},\mathfrak{g}_0)}$ is finite dimensional if and only if ${\mathcal{G}^{(0)}}$ is of finite type in the sense of Singer and Sternberg.

2.3. Examples

Example 1 Let ${N}$ be Heisenberg group. For the full prolongation, ${\mathcal{G}^{(0)}=\mathfrak{sl}(2,{\mathbb R})}$ contains rank one matrices, so the full prolongation is infinite dimensional.

Example 2 Let ${N}$ be Heisenberg group. Let ${\mathfrak{g}_0 =\{u\in Der_0 (\mathfrak{n})}$ whose restriction to ${\mathfrak{g}_{-1}}$ belongs to ${\mathfrak{co}(2)}$. Then ${\mathcal{G}^{(0)}=\mathfrak{so}(2)}$, so the prolongation is finite dimensional.

This implies that infinitesimal conformal contact transformations form a finite dimensional Lie algebra, but we have not proved it yet. The prolongation can be computed to be

$\displaystyle \begin{array}{rcl} Prol(\mathfrak{n},\mathfrak{g}_0)=\mathfrak{n}\oplus\mathfrak{g}_0\oplus\mathfrak{g}_1\oplus\mathfrak{g}_2 =\mathfrak{su}(2,1). \end{array}$

Example 3 Let ${N}$ be the group of unipotent upper triangular ${4\times 4}$-matrices. Then the full prolongation is

$\displaystyle \begin{array}{rcl} Prol(\mathfrak{n})=\mathfrak{n}\oplus\mathfrak{g}_0\oplus\mathfrak{g}_1\oplus\mathfrak{g}_2\oplus\mathfrak{g}_3. \end{array}$

Whereas ${\mathcal{G}^{(0)}(\mathfrak{n})}$ vanishes.

2.4. Prolongation of contact vectorfields

We are aiming at relating the Lie algebra of infinitesimal contact transformations to the full prolongation.

Definition 8 View a vectorfield on ${N}$ as a ${\mathfrak{n}}$-valued function on ${N}$, ${A_U : N\rightarrow\mathfrak{n}}$.

Note that ${A_{[U,W]}=[A_U,A_V]+U.A_W-W.A_U}$.

If ${U}$ is a contact vectorfield and ${W}$ is horizontal and left-invariant, then ${[U,W]}$ is horizontal again, and

$\displaystyle \begin{array}{rcl} A_{[U,W]}=[A_U,W]-W.A_U. \end{array}$

This implies that the right hand side is horizontal, i.e., for all ${m\geq 2}$,

$\displaystyle \begin{array}{rcl} W.A_{U}^{-m}=[A_U^{-m+1},W]. \end{array}$

This set of equations characterizes infinitesimal contact transformations.

Example 4 Let ${N}$ be the Heisenberg group. Then ${V=fX+gY+hT}$ is a contact vectorfield if and only if ${Xh=-g}$ and ${Yh=f}$.

Indeed, in this example,

$\displaystyle \begin{array}{rcl} A_{V}^{-1}=(f,g,0), \quad A_{V}^{-2}=(0,0,h), \end{array}$

$\displaystyle \begin{array}{rcl} Xh=X.A_{V}^{-2}=[A_{V}^{-1},X]=-(0,0,g), \quad \end{array}$

Iterating the equation above gives, for all ${m>r}$, and left-invariant ${W\in\mathfrak{g}_{-r}}$,

$\displaystyle \begin{array}{rcl} W_{-r}.A_{U}^{-m}=[A_U^{-m+1},W_{-r}]. \end{array}$

Proposition 9 Let ${U}$ be a vectorfield. The expression

$\displaystyle \begin{array}{rcl} A_U^0 (W_{-r})=W_{-r}.A_{U}^{-r},\quad W_{-r}\in\mathfrak{g}_{-r}, \end{array}$

defines a degree ${0}$ derivation of ${\mathfrak{n}}$. Therefore, it defines a map ${A_U^0 : N\rightarrow \mathfrak{g}_0}$.

Example 5 Heisenberg again. Then

$\displaystyle \begin{array}{rcl} A_V^0 =\begin{pmatrix} Xf &Yf&0 \\ Xg &Yg&0\\ 0&0&Th \end{pmatrix}. \end{array}$

The contact equations are expressible in terms of ${A_V}$, ${A_V^0 \in\mathfrak{g}}$, ${A_V^1}$, …. which are higher and higher derivatives. This provides us with a map from infinitesimal contact transformations to the full prolongation. One shows that this map is an isomorphism.

Alessandro Ottazzi’s own notes of this minicourse can be found here.