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Difference between revisions of "Infinitesimal structure"

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A structure on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i0509901.png" />-dimensional differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i0509902.png" /> that is determined by a reduction of the differentiable structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i0509903.png" /> of the principal bundle of frames of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i0509904.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i0509905.png" />, i.e. of invertible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i0509906.png" />-jets from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i0509907.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i0509908.png" /> with origin at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i0509909.png" />, to a certain Lie subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099010.png" /> of it. In other words, an infinitesimal structure of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099011.png" /> is given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099012.png" /> if a certain section is distinguished in the quotient bundle of the principal bundle of frames of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099014.png" /> by a Lie subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099015.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099016.png" /> an infinitesimal structure is also called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099018.png" />-structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099019.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099020.png" /> it is also called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099022.png" />-structure of higher order. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099023.png" /> is replaced by the projective differentiable group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099024.png" /> (a certain quotient group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099025.png" />), then the corresponding infinitesimal structure is called a projective infinitesimal structure.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099026.png" /> 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.
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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====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.F. Laptev,  "Fundamental infinitesimal structures of higher order on a smooth manifold"  ''Trudy Geom. Sem.'' , '''1'''  (1966)  pp. 139–189  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.S. Chern,  "The geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050990/i05099027.png" />-structures"  ''Bull. Amer. Math. Soc.'' , '''72''' :  2  (1966)  pp. 167–219</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.F. Laptev,  "Fundamental infinitesimal structures of higher order on a smooth manifold"  ''Trudy Geom. Sem.'' , '''1'''  (1966)  pp. 139–189  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.S. Chern,  "The geometry of $G$-structures"  ''Bull. Amer. Math. Soc.'' , '''72''' :  2  (1966)  pp. 167–219</TD></TR></table>
  
  

Latest revision as of 12:01, 5 July 2014

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


Comments

References

[a1] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)
How to Cite This Entry:
Infinitesimal structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infinitesimal_structure&oldid=32379
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article