The Hierarchy of Curvatures in Exceptional Geometry

Abstract: Despite remarkable success in describing supergravity reductions and backgrounds, generalized geometry and the closely related exceptional field theory are still lacking a fundamental object of differential geometry, the Riemann tensor. We explain that to construct such a tensor, an as of yet overlooked hierarchy of connections is required. They complement the spin connection with higher representations known from the tensor hierarchy of gauged supergravities. In addition to solving an important conceptual problem, this idea allows to define and explicitly construct generalized homogeneous spaces. They are the underlying structures of generalized U-duality, admit consistent truncations and provide a huge class of new backgrounds for flux compactifications with non-trivial generalized structure groups.