# Difference between revisions of "Infinitesimal structure"

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− | A structure on an | + | {{TEX|done}} |

+ | 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 | + | 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 | + | <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=17126