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- W1979149075 abstract "We generalize the usual definition of the Lie derivative to the case of a morphism between fibered manifolds which does not necessarily preserve the base. We prove that the vanishing of the Lie derivatives is a necessary and sufficient condition for the equivariance of a morphism of fibered manifolds under the action of a connected Lie group. 0. Introduction. In many problems in mathematical physics one considers a 'field', in effect, a section of a fibered manifold (which is very often a vector bundle) over a base manifold representing space-time, and a group of transformations of the base. To know whether the field is physically meaningful, one asks the classical question, whether the group of transformations is a 'symmetry group' (in alternate terminology, an 'invariance group') of the field, that is, does there exist a lifting of the action of the group on the base manifold to an action on the fibered manifold by automorphisms (if the fibered manifold is a vector bundle one generally requires that the group act by vector-bundle automorphisms) such that the section under consideration be invariant under that action of the group? More generally, one can ask the same question regarding a differential operator from one fibered manifold to another over the same base manifold. One must consider liftings of the group action to the two fibered manifolds: if there exist liftings such that the differential operator is equivariant with respect to the two lifted actions, the group is called a 'symmetry group' (or an 'invariance group') of the differential operator or of the associated system of partial differential equations. (For scalar differential operators or, more generally, for differential operators on tensor bundles, the word 'invariant' is usually reserved for 'invariant under the natural liftings' while the word 'covariant' can be used in the other cases.) In general, one assumes that the group is a Lie group, considers its corresponding infinitesimal transformations, and determines whether the section or the differential operator is 'infinitesimally invariant'. In the case of a section that means, does there exist a lifting of the Lie algebra of vector fields on the base to a Lie algebra of infinitesimal automorphisms of the fibered manifold such that the section is infinitesimally invariant, i.e., such that its Lie derivatives with respect to those vector fields defined by this lifting vanish? Received by the editors March 6, 1978 and, in revised form, June 13, 1978. AMS (MOS) subject classifications (1970). Primary 53A55; Secondary 50A15, 22A60." @default.
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- W1979149075 date "1979-03-01" @default.
- W1979149075 modified "2023-09-25" @default.
- W1979149075 title "Infinitesimal conditions for the equivariance of morphisms of fibered manifolds" @default.
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- W1979149075 doi "https://doi.org/10.1090/s0002-9939-1979-0545599-9" @default.
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