Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have " K " v-rational points for all places " v " of " K ".
42.
A principal homogeneous space of is a manifold abstractly characterized by having a exponential map of the Lie algebra and in this way obtain, locally, a group action on.
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In accordance with the Erlangen program, the geometry of Minkowski space is defined by the Poincar?group : Minkowski space is considered as a homogeneous space for the group.
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The problem is then to show that is a smooth function near ( since topological groups are homogeneous spaces, they look the same everywhere as they do near ).
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The subgroups for which the corresponding homogeneous space has an invariant complex structure correspond to parabolic subgroups in the complexification of the compact Lie group, a reductive algebraic group.
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This then specializes to the case when the specified fibre is a principal homogeneous space for the left action of the group on itself, yielding the associated principal bundle.
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The study of the properties of unipotent and quasiunipotent flows on homogeneous spaces remains an active area of research, with applications to further questions in the theory of Diophantine approximation.
48.
In other words, there is locally a diffeomorphism of into the homogeneous space, such that is the pullback of the Maurer & ndash; Cartan form along some section of the tautological bundle.
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Another line of development in which p . d . kernels played a large role was the theory of harmonics on homogeneous spaces as begun by unitary representations of locally compact groups.
50.
The geodesics emanating from the origin in the homogeneous spaces can be identified with one parameter groups with generators in i \ mathfrak { m } or \ mathfrak { m }.
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