Cartan Geometries and their Symmetries by Mike Crampin & David Saunders

Author:Mike Crampin & David Saunders
Language: eng
Format: epub
Publisher: Atlantis Press, Paris

Thus is projection of to along . Now under the assumption that , the restriction of to is injective; and then since it is an isomorphism . That is to say, the nondegeneracy condition on the infinitesimal connection in the definition of an infinitesimal Cartan geometry can be stated equivalently as the requirement that the kernel projection is such that for each the linear map restricts to an isomorphism .

Let us interpret this result in terms of the representation of elements of as projectable vector fields along fibres of . For an infinitesimal Cartan geometry we have a global section , and consists of those elements of which, considered as vector fields along , are tangent to the submanifold . Moreover, the elements of , considered as vector fields tangent to , vanish at . But at each point p of , (the tangent space to the fibre, that is, the vertical subspace of ) is isomorphic to ; so in particular the vector space spanned by the values of the elements of at is the whole of . Now induces an isomorphism of with ; on restriction to this induces an isomorphism of with . But is just an isomorphic copy of . We have shown that in any Cartan geometry the kernel projection induces for each an isomorphism . That is to say, for any Cartan geometry the bundle E is soldered to M along the image of the section , the soldering being defined by the kernel projection of the infinitesimal connection.

We examine this construction in more detail.

Let be a basis for , with a basis for . Take a local trivialization of over , with corresponding vertical vector fields on forming a basis for . Then vanishes on .

Any element may be written as

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