Infinitesimal structure

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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.


[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



[a1] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)
How to Cite This Entry:
Infinitesimal structure. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article