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.


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