# Differential-geometric structure

One of the fundamental concepts in modern differential geometry including the specific structures studied in classical differential geometry. It is defined for a given differentiable manifold $M ^ {n}$ as a differentiable section in a fibre space $( X _ {F} , p _ {F} , M ^ {n} )$ with base $M ^ {n}$ associated with a certain principal bundle $( X , p , M ^ {n} )$ or, according to another terminology, as a differentiable field of geometric objects on $M ^ {n}$. Here $F$ is some differentiable $\mathfrak G$- space where $\mathfrak G$ is the structure Lie group of the principal bundle $( X , p , M ^ {n} )$ or, in another terminology, the representation space of the Lie group $\mathfrak G$.

If $( X , p , M ^ {n} )$ is the principal bundle of frames in the tangent space to $M ^ {n}$, $G$ is some closed subgroup in $\mathfrak G = \mathop{\rm GL} ( n, \mathbf R )$, and $F$ is the homogeneous space $\mathfrak G / G$, the corresponding differential-geometric structure on $M ^ {n}$ is called a $G$- structure or an infinitesimal structure of the first order. For example, if $G$ consists of those linear transformations (elements of $\mathop{\rm GL} ( n , \mathbf R )$) which leave an $m$- dimensional space in $\mathbf R ^ {n}$ invariant, the corresponding $G$- structure defines a distribution of $m$- dimensional subspaces on $M ^ {n}$. If $G$ is the orthogonal group $O ( n , \mathbf R )$— the subgroup of elements of $\mathop{\rm GL} ( n , \mathbf R )$ which preserve the scalar product in $\mathbf R ^ {n}$—, then the $G$- structure is a Riemannian metric on $M ^ {n}$, i.e. the field of a positive-definite symmetric tensor $g _ {ij}$. In a similar manner, almost-complex and complex structures are special cases of $G$- structures on $M ^ {n}$. A generalization of the concept of a $G$- structure is an infinitesimal structure of order $r$, $r > 1$( or $G$- structure of a higher order); here $( X , p , M ^ {n} )$ is the principal bundle of frames of the order $r$ on $M ^ {n}$, and $G$ is a closed subgroup of its structure group $D _ {n} ^ {r}$.

All kinds of connections (cf. Connection) are important special cases of differential-geometric structures. For instance, a connection in a principal bundle is obtained if the role of $M ^ {n}$ is played by the space $P$ of some principal bundle $( P , p , B )$, and the $G$- structure on $P$ is the distribution of $m$- dimensional, $m = \mathop{\rm dim} P - \mathop{\rm dim} B$, subspaces complementary to the tangent spaces of the fibres which is invariant with respect to the action on $P$ of the structure group of the bundle. Connections on a manifold $M ^ {n}$ are special cases of differential-geometric structures on $M ^ {n}$, but more general ones than $G$- structures on $M ^ {n}$. For instance, an affine connection on $M ^ {n}$, definable by a field of connection objects $\Gamma _ {ij} ^ {k} ( x)$, is obtained as the differential-geometric structure on $M ^ {n}$ for which $( X , p , M ^ {n} )$ is the principal bundle of frames of second order, $\mathfrak G$ is its structure group $D _ {n} ^ {2}$, and the representation space $F$ of $D _ {n} ^ {2}$ is the space $\mathbf R ^ {3n}$ with coordinates $\Gamma _ {ij} ^ {k}$, where the representation is defined by the formulas

$$\overline \Gamma \; {} _ {st} ^ {r} = ( A _ {s} ^ {i} A _ {t} ^ {j} \Gamma _ {ij} ^ {k} + A _ {st} ^ {k} ) \overline{A}\; {} _ {k} ^ {r} ,$$

where

$$A _ {k} ^ {r} = \left ( \frac{\partial x ^ {r} }{\partial \overline{x}\; {} ^ {k} } \right ) _ {0} ,\ A _ {st} ^ {k} = \ \left ( \frac{\partial ^ {2} x ^ {k} }{\partial \overline{x}\; {} ^ {s} \partial x bar {} ^ {t} } \right ) _ {0}$$

are the coordinates of an element of the group $D _ {n} ^ {2}$, and $A _ {k} ^ {r} \overline{A}\; {} _ {t} ^ {k} = \delta _ {t} ^ {r}$. In the case of a projective connection on $M ^ {n}$ one deals with a certain representation of $D _ {n} ^ {3}$ in $\mathbf R ^ {3 ( n+ 1 ) }$, while in cases of connections of a higher order, one deals with representations of $D _ {n} ^ {r}$. By this approach the theory of differential-geometric structures becomes closely related to the theory of geometric objects (Cf. Geometric objects, theory of).

#### References

 [1] O. Veblen, J.H.C. Whitehead, "The foundations of differential geometry" , Cambridge Univ. Press (1932) (Appendix by V.V. Vagner in the Russian translation) [2] G.F. Laptev, "Differential geometry of imbedded manifolds. Group-theoretical method of differential geometric investigation" Trudy Moskov. Mat. Obshch. , 2 (1953) pp. 275–382 (In Russian) [3] D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) [4] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)