on a manifold
A principal subbundle with structure group of the principal bundle of co-frames on the manifold. More exactly, let be the principal -bundle of all co-frames of order over an -dimensional manifold , and let be a subgroup of the general linear group of order . A submanifold of the manifold of -co-frames defines a -structure of order , , if defines a principal -bundle, i.e. the fibres of are orbits of . For example, a section of (a field of co-frames) defines a -structure , which is called the -structure generated by this field of co-frames. Any -structure is locally generated by a field of co-frames. The -structure over the space generated by the field of co-frames , where is the identity mapping, is called the standard flat -structure.
Let be a -structure. The mapping of the manifold into the point can be extended to a -equivariant mapping , which can be considered as a structure of type on . If the homogeneous space is imbedded as an orbit in a vector space admitting a linear action of , then the structure can be considered as a linear structure of type ; this is called the Bernard tensor of the -structure , and is often identified with it. Conversely, let be a linear geometric structure of type (for example, a tensor field), whereby belongs to a single orbit of . is then a -structure, where is the stabilizer of the point in , and is its Bernard tensor. For example, a Riemannian metric defines an -structure, an almost-symplectic structure defines a -structure, an almost-complex structure defines a -structure, and a torsion-free connection defines a -structure of the second order ( is considered here as a subgroup of the group ). An affinor (a field of endomorphisms) defines a -structure if and only if it has at all points one and the same Jordan normal form , where is the centralizer of the matrix in .
The elements of the manifold can be considered as co-frames of order 1 on , which makes it possible to consider the natural bundle as an -structure of order one, where is the kernel of the natural homomorphism . Every -structure of order has a related sequence of -structures of order one,
where . Consequently, the study of -structures of higher order reduces to the study of -structures of order one. A co-frame can be considered as an isomorphism .
The -form , assigning to a vector the value , is called the displacement form. In the local coordinates of , the form is expressed as , where is the standard basis in .
The restriction of on a -structure is called the displacement form of the -structure. It possesses the following properties: 1) strong horizontality: ; and 2) -equivariance: for any .
Using the form it is possible to characterize the principal bundles with base that are isomorphic to a -structure. Namely, a principal -bundle is isomorphic to a -structure if and only if there are a faithful linear representation of the group in an -dimensional vector space , , and a -valued strongly-horizontal -equivariant -form on . Removal of the requirement that the representation be faithful gives the concept of a generalized -structure (of order one) on , namely a principal -bundle with a linear representation , , and a -valued strongly-horizontal -equivariant -form on .
An example of a generalized -structure is the canonical bundle over the homogeneous space of a Lie group . Here is the isotropy representation of the group , while is defined by the Maurer–Cartan form of .
Let be a -structure of order one. The bundle of -jets of local sections of can be considered as a -structure on , where is a commutative group, is the Lie algebra of , that is linearly represented in the space by the formula
and that acts on the manifold according to the formula
where is the canonical isomorphism of the Lie algebra of the group onto the vertical subspace . Here the element is considered as a horizontal (i.e. complementary to the vertical) subspace in . It defines a co-frame , which is defined on a vertical subspace by the mapping , and on a horizontal subspace by the mapping . The vector function , defined by the formula , , is called the torsion function of the -structure . A section of the bundle defines a connection on , while the restriction of the function on is a function defining the coordinates of the torsion tensor of this connection relative to the field of co-frames .
The mapping is -equivariant relative to the above-mentioned action of on and to the action of on , which is defined by the formula
where , . The mapping induced by the mapping is called the structure function of the -structure , the vanishing of is equivalent to the existence of a torsion-free connection on .
The choice of a subspace complementary to defines a subbundle of the bundle of co-frames with structure group , i.e. a -structure on . It is called the first prolongation of the -structure . The -th prolongation is defined by induction as the -structure on , where the group is isomorphic to the vector group . The structure function of the -th prolongation is called the structure function of -th order of the -structure .
The central problem of the theory of -structures is the local equivalence problem, i.e. the problem of finding necessary and sufficient conditions under which two -structures and with the same structure group are locally equivalent, i.e. a local diffeomorphism of the manifolds and should exist that induces an isomorphism of -structures over the neighbourhoods and . A particular case of this problem is the integrability problem, i.e. the problem of finding necessary and sufficient conditions for the local equivalence of a given -structure and the standard flat -structure. The local equivalence problem can be reformulated as the problem of finding a complete system of local invariants of a -structure.
For an -structure, which is identified with a Riemannian metric, the integrability problem was solved by B. Riemann: Necessary and sufficient conditions for integrability consist in the vanishing of the curvature tensor of the metric. The local equivalence problem was solved by E. Christoffel and R. Lipschitz: A complete system of local invariants of a Riemannian metric consists of its curvature tensor and its successive covariant derivatives (see ).
An approach to solving the equivalence problem is based on the concepts of a prolongation and a structure function. Every -structure of order one with structure group is connected with a sequence of prolongations
and a sequence of structure functions . For an -structure, the structure function on is equal to 0, while the essential parts of the remaining structure functions , , are identified with the curvature tensor of the corresponding metric and its successive covariant derivatives. For to be integrable it is necessary and sufficient that the structure functions be constant, and that their values coincide with the corresponding values of the structure functions of the standard flat -structure (see , , ). The number depends only on the group . For a broad class of linear groups, especially for all irreducible groups that do not belong to Berger's list of holonomy groups of spaces with a torsion-free affine connection , one has , and for a -structure to be integrable it is necessary and sufficient that the structure function vanishes, or that a torsion-free linear connection exists, preserving the -structure.
A -structure is called a -structure of finite type (equal to ) if , . In this case is a field of co-frames (an absolute parallelism), and the automorphism group of the -structure is isomorphic to the automorphism group of this parallelism and is a Lie group. The local equivalence problem of these structures reduces to the equivalence problem of absolute parallelisms and has been solved in terms of a finite sequence of structure functions (see ). For a -structure of infinite type, the local equivalence problem remains unsolved in the general case (1984).
Two -structures and are called formally equivalent at the points , if an isomorphism of the fibres exists that can be continued to an isomorphism of the corresponding fibres of the prolongations and . Examples have been found which demonstrate that if two -structures of class are formally equivalent for all pairs , then it does not follow, generally speaking, that they are locally equivalent . In the analytic case, proper subsets , exist, which are countable unions of analytic sets, such that for any , , the formal equivalence of two structures and at the points implies that they are locally equivalent .
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