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For the geometry of oriented (2, 3, 5) distributions (M,D), which correspond to regular, normal parabolic geometries of type (G2, P) for a particular parabolic subgroup P < G2, we develop the corresponding tractor calculus and use it to analyze the first BGG operator ⴱ0 associated to the 7-dimensional irreducible representation of G2. We give an explicit formula for the normal connection on the corresponding tractor bundle and use it to derive explicit expressions for this operator. We also show that solutions of this operator are automatically normal, yielding a geometric interpretation of ker ⴱ0: For any (M,D), this kernel consists precisely of the almost Einstein scales of the Nurowski conformal structure on M that D determines. We apply our formula for ⴱ0 (1) to recover efficiently some known solutions, (2) to construct a distribution with root type [3, 1] with a nonzero solution, and (3) to show efficiently that the conformal holonomy of a particular (2, 3, 5) conformal structure is equal to G2. [ABSTRACT FROM AUTHOR]

The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing representation. Consequences for homogeneous models and conformal holonomy are described. A careful presentation is made of background material concerning standard tractor bundles and equivalence between parabolic geometries and underlying structures. [ABSTRACT FROM AUTHOR]