# Infinitesimal structure

A structure on an -dimensional differentiable manifold that is determined by a reduction of the differentiable structure group of the principal bundle of frames of order on , i.e. of invertible -jets from to with origin at , to a certain Lie subgroup of it. In other words, an infinitesimal structure of order is given on if a certain section is distinguished in the quotient bundle of the principal bundle of frames of order on by a Lie subgroup . For an infinitesimal structure is also called a -structure on , and for it is also called a -structure of higher order. If is replaced by the projective differentiable group (a certain quotient group of ), 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 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 -structures" Bull. Amer. Math. Soc. , 72 : 2 (1966) pp. 167–219 |

#### Comments

#### References

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

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Infinitesimal structure.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Infinitesimal_structure&oldid=17126