Carnot group

A Carnot (or stratified) group of step ${\displaystyle k}$ is a connected, simply connected, finite-dimensional Lie group whose Lie algebra ${\displaystyle {\mathfrak {g}}}$ admits a step-${\displaystyle k}$ stratification. Namely, there exist nontrivial linear subspaces ${\displaystyle V_{1},\cdots ,V_{k}}$ such that

${\displaystyle {\mathfrak {g}}=V_{1}\oplus \cdots \oplus V_{k}}$, ${\displaystyle [V_{1},V_{i}]=V_{i+1}}$ for ${\displaystyle i=1,\cdots ,k-1}$, and ${\displaystyle [V_{1},V_{k}]=\{0\}}$.

Note that this definition implies the first stratum ${\displaystyle V_{1}}$ generates the whole Lie algebra ${\displaystyle {\mathfrak {g}}}$.

The exponential map is a diffeomorphism from ${\displaystyle {\mathfrak {g}}}$ onto ${\displaystyle G}$. Using these exponential coordinates, we can identify ${\displaystyle G}$ with ${\displaystyle (\mathbb {R} ^{n},\star )}$, where ${\displaystyle n=\dim V_{1}+\cdots +\dim V_{k}}$ and the operation ${\displaystyle \star }$ is given by the Baker–Campbell–Hausdorff formula.

Sometimes it is more convenient to write an element ${\displaystyle z\in G}$ as

${\displaystyle z=(z_{1},\cdots ,z_{k})}$ with ${\displaystyle z_{i}\in \mathbb {R} ^{\dim V_{i}}}$ for ${\displaystyle i=1,\cdots ,k}$.

The reason is that ${\displaystyle G}$ has an intrinsic dilation operation ${\displaystyle \delta _{\lambda }:G\to G}$ given by

${\displaystyle \delta _{\lambda }(z_{1},\cdots ,z_{k}):=(\lambda z_{1},\cdots ,\lambda ^{k}z_{k})}$.

The real Heisenberg group is a Carnot group which can be viewed as a flat model in Sub-Riemannian geometry as Euclidean space in Riemannian geometry. The Engel group is also a Carnot group.