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A $G$-structure $\pi : B_G \rightarrow M$ is said to be of $finite$ $type$ if $\mathfrak{g}^{(k)} = 0$ for some $k \in \mathbb{N}$, where $\mathfrak{g}^{(k)}$ denotes the $k$th prolongation of the Lie algebra $\mathfrak{g} = T_{e}G$. For finite type $G$-structures, let us call the first $k$ for which $\mathfrak{g}^{(k)} = 0$ the $order$ of the $G$-structure.

For example, :
• $O(n)$-structures (Riemannian metrics) are of finite type and order $1$, because $\mathfrak{o} (n)^{(1)} = 0$, but .
• But $Sp(n)$-structures (symplectic structures) are not of finite type because the group of symplectomorphisms is infinite dimensional.
• It can also be shown that $CO(n)$-structures (conformal structures) are of finite type and order $2$ (except if the dimension is $n > 2$)2\$, in which case it is not of finite type).

Are there any finite type $G$-structures of order greater than $2$?

More generally, are there $G$-structures of any order?

Thanks.

1

$G$-structures of finite type.

A $G$-structure $\pi : B_G \rightarrow M$ is said to be of $finite$ $type$ if $\mathfrak{g}^{(k)} = 0$ for some $k \in \mathbb{N}$, where $\mathfrak{g}^{(k)}$ denotes the $k$th prolongation of the Lie algebra $\mathfrak{g} = T_{e}G$. For finite type $G$-structures, let us call the first $k$ for which $\mathfrak{g}^{(k)} = 0$ the $order$ of the $G$-structure. For example, $O(n)$-structures (Riemannian metrics) are of finite type and order $1$, because $\mathfrak{o} (n)^{(1)} = 0$, but $Sp(n)$-structures are not of finite type. It can also be shown that $CO(n)$-structures are of finite type and order $2$ (if $n > 2$).

Are there any $G$-structures of order greater than $2$?

More generally, are there $G$-structures of any order?

Thanks.
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