The proposed project systematically investigates, using techniques from spin geometry, the structure of Riemannian symmetric and 3-symmetric spaces. These spaces, which arise naturally in many contexts within differential geometry, are characterized by the existence of a symmetry (of order 2 and 3 respectively) acting on the tangent space at each point. The existence of such symmetries imposes a rigid algebraic structure on these spaces, allowing difficult geometric considerations (e.g. curvature properties and the existence of globally defined geometric structures) to be reduced to simpler algebraic problems. These, in turn, may be approached using powerful classical results from Lie theory and representation theory.
Riemannian symmetric spaces in particular have been extremely thoroughly studied, including a full classification by Cartan nearly a century ago, yet despite this their spinorial properties remain largely unknown. Riemannian 3-symmetric spaces, which were introduced by Gray almost fifty years later, have received less attention than their symmetric counterparts, but nevertheless are prime candidates for study using spinorial techniques. The proposed project is threefold: firstly, to obtain a purely spinorial description of Hermitian symmetric spaces; secondly, to generalize this description to Riemannian 3-symmetric spaces; and thirdly, to study the beautiful class of examples given by the twistor spaces over quaternion Kahler symmetric spaces.

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