## Notes of Alessandro Ottazzi’s lecture nr 1

Mini course on Tanaka prolongation

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

Extends Singer and Sternberg’s theory (late sixties). Its goal is to understand the equivalence problem and deformations of ${G}$-structures. In the seventies, Tanaka adapted it to subRiemannian geometry. Nowadays, it can be used to solve problems in subRiemannian geometry like

• Determine the isometry group of a subRiemannian manifold.
• Determine the conformal group of a subRiemannian manifold.
• Guess differential invariants of geometric structures on plane distributions.

We shall explain the theory, starting from the classical case, and show some applications.

Plan

1. ${G}$-structures
2. Singer-Sternberg prolongation
3. The Liouville theorem for conformal maps of ${{\mathbb R}^n}$
4. Tanaka prolongation theory
5. Liouville theorem for Carnot groups

Reference for the first 3 sections: Kobayashi’s book Transformation groups. The original paper by Singer and Sternberg contains a historical overview of the problem.

1. ${G}$-structures

Let ${M}$ be a smooth ${n}$-manifold. Let ${G}$ be a Lie group.

Definition 1 A ${G}$-principal bundle over ${M}$ is a manifold ${P}$ with a free right ${G}$-actions and a submersion ${P\rightarrow M}$ which induces a diffeomorphism ${P/G \simeq M}$.

Definition 2 The frame bundle ${LM}$ over ${M}$ is the union of ${L_m M}$, ${m\in M}$, where ${L_m M}$ is the set of ordered bases of ${T_m M}$. It admits a manifold structure which turns it into a ${Gl(n,{\mathbb R})}$-principal bundle.

Definition 3 Let ${G\subset Gl(n,{\mathbb R})}$ be a closed subgroup. A ${G}$-structure on ${M}$ is a ${G}$-sub-bundle of the frame bundle ${LM}$ over ${M}$, i.e. ${P\subset LM}$, and for ${p\in P}$ and ${A\in Gl(n,{\mathbb R})}$, ${pA\in P \Leftrightarrow A\in G}$.

Definition 4 A ${G}$-structure ${P}$ on ${M}$ is integrable if locally there exist local coordinates whose frames belongs to ${P}$.

Example 1 If ${G}$ is trivial, a ${\{1\}}$-structure is a global frame. Such a structure exists if and only if ${M}$ is parallelizable.

Definition 5 Let ${P}$ be a ${G}$-structure on ${M}$. An automorphism of ${P}$ is a diffeomorphism ${f:M\rightarrow M}$ such that the induced tangent map ${Tf(P)\subset P}$. Equivalently, when expressed in frames taken from ${P}$, the differential of ${f}$ at each point belongs to ${G}$.

Definition 6 Let ${P}$ be a ${G}$-structure on ${M}$. An infinitesimal automorphism of ${P}$ is a vectorfield whose local flow is made of automorphisms.

Remark 1 If the ${G}$-structure is integrable, a vectorfield ${V}$ is an infinitesimal automorphism if and only if, in adapted coordinates, the differential of ${V}$ belongs to the Lie algebra ${\mathfrak{g}}$.

Example 2 An ${O(n)}$-structure is the same as a Riemannian metric. Automorphisms are isometries, infinitesimal automorphisms are called Killing fields. An ${O(n)}$-structure is integrable if and only if it is flat.

Example 3 An ${Sl(n,{\mathbb R})}$-structure is the same as a choice of orientation and volume form. Automorphisms are volume and orientation preserving diffeomorphisms. Infinitesimal automorphisms are divergence free vectorfields. Every ${Sl(n,{\mathbb R})}$-structure is integrable.

This is an example of a ${G}$-structure of infinite type: automorphisms groupes are infinite dimensional.

2. Singer and Sternberg theory

We work locally, in ${{\mathbb R}^n}$, and with the flat ${G}$-structure. Let ${V=\sum_{i=1}^{n}v_i \frac{\partial}{\partial x_i}}$ be a vectorfield. We want to characterize infinitesimal automorphisms in terms of their Taylor expansions. Write

$\displaystyle \begin{array}{rcl} v_i(x)=v_i(0)+\sum_{k=1}^{\infty}\frac{1}{k!}a^{i}_{j_1,...,j_k}x_{j_1}\cdots x_{j_k}, \end{array}$

where

$\displaystyle \begin{array}{rcl} a^{i}_{j_1,...,j_k}=\frac{\partial^k}{\partial x_{j_1}...\partial x_{j_k}}(0). \end{array}$

We note that

• The ${a^{i}_{j_1,...,j_k}}$ are symmetric in the lower indices.
• The matrix ${(\frac{\partial v^i}{\partial x_j})_{ij}\in\mathfrak{g}}$.
• Matrices ${(a^{i}_{j_1...j_k})_{ij_1}\in\mathfrak{g}}$.

2.1. Formal definition of Singer-Sternberg prolongation

Definition 7 For ${k\in{\mathbb N}}$, let ${\mathcal{G}^{(k)}(\mathfrak{g})}$ denote the vectorspace of symmetric multilinear maps ${T:{\mathbb R}^n \times\cdots\times{\mathbb R}^n \rightarrow {\mathbb R}^n}$ such that for all ${v_1,\ldots,v_k\in{\mathbb R}^n}$, ${v\mapsto T(v,v_1,\ldots,v_k)\in\mathfrak{g}}$.

In fact, it is an inductive definition: ${T\in \mathcal{G}^{(k)}(\mathfrak{g})}$ if and only if for all ${v\in{\mathbb R}^n}$, ${(v_1,\ldots,v_k)\mapsto T(v,v_1,\ldots,v_k)\in\mathcal{G}^{(k-1)}(\mathfrak{g})}$, and ${T}$ is symmetric.

Definition 8 ${\mathfrak{g}}$ is of type ${k}$ if ${k}$ is the smallest integer such that ${\mathcal{G}^{(k)}(\mathfrak{g})=0}$. It is of infinite type if no such ${k}$ exists.

2.2. The Lie algebra of jets

Denote by ${\mathcal{G}^{(-1)}(\mathfrak{g})={\mathbb R}^n}$. We define a graded Lie algebra structure on ${\bigoplus_{k\geq -1}\mathcal{G}^{(k)}(\mathfrak{g})}$ which reflects the Lie algebra structure on the space of infinitesimal automorphisms of the integrable ${G}$-structure ${P}$. If ${T\in \mathcal{G}^{(k)}(\mathfrak{g})}$ and ${T\in\mathcal{G}^{(k')}(\mathfrak{g})}$, let

$\displaystyle \begin{array}{rcl} [T,T'](v_0,\ldots,v_{k+k'})&=&\frac{1}{k!(k'+1)!}\sum T(T'(v_{j_1},\ldots,v_{j_k'}),v_{j_{k'+1}},\ldots,v_{j_{k+k'}})\\ &&-\frac{1}{(k+1)!k'!}\sum T'(T(v_{j_1},\ldots,v_{j_k}),v_{j_{k+1}},\ldots,v_{j_{k+k'}}), \end{array}$

where one sums over permutations. If ${T\in\mathcal{G}^{(k)}(\mathfrak{g})}$ and ${v\in\mathcal{G}^{(-1)}(\mathfrak{g})={\mathbb R}^n}$,

$\displaystyle \begin{array}{rcl} [T,v](v_1,\ldots,v_k)=T(v,v_1,\ldots,v_k). \end{array}$

By construction, the polynomial vectorfields ${V=T(x,\ldots,x)}$ and ${V'=T'(x,\ldots,x)}$ are infinitesimal automorphisms, their bracket is ${[V,V']=[T,T'](x,\ldots,x)}$.

2.3. Examples

Example 4 ${G=O(n)}$. Then the prolongation is ${{\mathbb R}^n\oplus\mathfrak{so}(n)}$.

Indeed, ${\mathcal{G}^{(0)}(\mathfrak{so}(n))}$ consists of skew symmetric matrices, ${\mathcal{G}^{(1)}(\mathfrak{so}(n))}$ consists of ${(2,1)}$-tensors which are symmetric in two indices and skew-symmetric in two others. A classical lemma asserts that such tensors vanish. Indeed,

$\displaystyle \begin{array}{rcl} T^{i}_{jk}=-T^{k}_{ji}=-T^{k}_{ij}=T^{j}_{ik}=T^{j}_{ki}=-T^{i}_{kj}=-T^{i}_{jk}. \end{array}$

which implies that ${T=0}$.

Example 5 ${G=Sl(n,{\mathbb R})}$. Then the prolongation is infinite dimensional.

Indeed, it is always so for Lie algebras which contain a matrix of rank ${1}$. To prove this, assume ${A=v\otimes w^* \in\mathfrak{g}}$. Let ${k>0}$. Set

$\displaystyle \begin{array}{rcl} T(v_1,\ldots,v_{k+1})=w(v_1)\ldots w(v_{k+1})v. \end{array}$

Moreover,

$\displaystyle \begin{array}{rcl} z\mapsto T(v_1,\ldots,v_{k},z)=w(v_1)\ldots w(v_{k})Az \end{array}$

belongs to ${\mathfrak{g}}$. Thus ${T\in \mathcal{G}^{(k)}(\mathfrak{g})}$.