Infinitesimal structure

A structure on an $n$-dimensional differentiable manifold $M^n$ that is determined by a reduction of the differentiable structure group $D_n^r$ of the principal bundle of frames of order $r$ on $M^n$, i.e. of invertible $r$-jets from $\mathbf R^n$ to $M^n$ with origin at $0\in\mathbf R^n$, to a certain Lie subgroup $G$ of it. In other words, an infinitesimal structure of order $r$ is given on $M^n$ if a certain section is distinguished in the quotient bundle of the principal bundle of frames of order $r$ on $M^n$ by a Lie subgroup $G\subset D_n^r$. For $r=1$ an infinitesimal structure is also called a $G$-structure on $M^n$, and for $r>1$ it is also called a $G$-structure of higher order. If $D_n^r$ is replaced by the projective differentiable group $PD_n^r$ (a certain quotient group of $D_n^{r+1}$), then the corresponding infinitesimal structure is called a projective infinitesimal structure.

The structure equations are a tool for studying infinitesimal structures. The basic problems in the study of infinitesimal structures are: finding topological characteristics of a manifold $M^n$ having a certain infinitesimal structure, distinguishing the infinitesimal structures that are extensions of some infinitesimal structure of lower order, the problem of integrability of an infinitesimal structure, etc.

References

[1]	G.F. Laptev, "Fundamental infinitesimal structures of higher order on a smooth manifold" Trudy Geom. Sem. , 1 (1966) pp. 139–189 (In Russian)
[2]	S.S. Chern, "The geometry of $G$-structures" Bull. Amer. Math. Soc. , 72 : 2 (1966) pp. 167–219

References

[a1]	S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)