** 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 -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 **

- -structures
- Singer-Sternberg prolongation
- The Liouville theorem for conformal maps of
- Tanaka prolongation theory
- 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. -structures **

Let be a smooth -manifold. Let be a Lie group.

Definition 1A -principal bundle over is a manifold with a free right -actions and a submersion which induces a diffeomorphism .

Definition 2The frame bundle over is the union of , , where is the set of ordered bases of . It admits a manifold structure which turns it into a -principal bundle.

Definition 3Let be a closed subgroup. A -structure on is a -sub-bundle of the frame bundle over , i.e. , and for and , .

Definition 4A -structure on is integrable if locally there exist local coordinates whose frames belongs to .

Example 1If is trivial, a -structure is a global frame. Such a structure exists if and only if is parallelizable.

Definition 5Let be a -structure on . An automorphism of is a diffeomorphism such that the induced tangent map . Equivalently, when expressed in frames taken from , the differential of at each point belongs to .

Definition 6Let be a -structure on . An infinitesimal automorphism of is a vectorfield whose local flow is made of automorphisms.

Remark 1If the -structure is integrable, a vectorfield is an infinitesimal automorphism if and only if, in adapted coordinates, the differential of belongs to the Lie algebra .

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

Example 3An -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 -structure is integrable.

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

**2. Singer and Sternberg theory **

We work locally, in , and with the flat -structure. Let be a vectorfield. We want to characterize infinitesimal automorphisms in terms of their Taylor expansions. Write

where

We note that

- The are symmetric in the lower indices.
- The matrix .
- Matrices .

** 2.1. Formal definition of Singer-Sternberg prolongation **

Definition 7For , let denote the vectorspace of symmetric multilinear maps such that for all , .

In fact, it is an inductive definition: if and only if for all , , and is symmetric.

Definition 8is of type if is the smallest integer such that . It is of infinite type if no such exists.

** 2.2. The Lie algebra of jets **

Denote by . We define a graded Lie algebra structure on which reflects the Lie algebra structure on the space of infinitesimal automorphisms of the integrable -structure . If and , let

where one sums over permutations. If and ,

By construction, the polynomial vectorfields and are infinitesimal automorphisms, their bracket is .

** 2.3. Examples **

Example 4. Then the prolongation is .

Indeed, consists of skew symmetric matrices, consists of -tensors which are symmetric in two indices and skew-symmetric in two others. A classical lemma asserts that such tensors vanish. Indeed,

which implies that .

Example 5. Then the prolongation is infinite dimensional.

Indeed, it is always so for Lie algebras which contain a matrix of rank . To prove this, assume . Let . Set

Moreover,

belongs to . Thus .