  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.
