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)

References

[a1]	M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish (1972–1975) pp. 1–5