Explicit Sp(2,R)-invariant split G2 structures are constructed on two homogeneous spaces, with tau2=0 in the first family and tau1=tau2=0 in the second.
Aerodynamics of flying saucers
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abstract
We identify various structures on the configuration space C of a flying saucer, moving in a three-dimensional smooth manifold M. Always C is a five-dimensional contact manifold. If M has a projective structure, then C is its twistor space and is equipped with an almost contact Legendrean structure. Instead, if M has a conformal structure, then the saucer moves according to a CR structure on C. With yet another structure on M, the contact distribution in C is equipped with a cone over a twisted cubic. This defines a certain type of Cartan geometry on C (more specifically, a type of `parabolic geometry') and we provide examples when this geometry is `flat,' meaning that its symmetries comprise the split form of the exceptional Lie algebra G2.
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On certain classes of $Sp(2,R)$ symmetric $G_2$ structures
Explicit Sp(2,R)-invariant split G2 structures are constructed on two homogeneous spaces, with tau2=0 in the first family and tau1=tau2=0 in the second.