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Alexandrov geometry studies non smooth analogues of Riemannian manifolds with curvature bounded from below or above. It includes spaces with curvature bounded below (briefly $\mathrm{CBB}[\kappa]$) and spaces with curvature bounded above (briefly $\mathrm{CAT}[\kappa]$).

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Embedding expanders in CAT(0) spaces

Here is a class of Hadamard spaces in which, I think, no expander embeds: Consider a Hadamard space $X$, which is also "FDSCBB" in the sense Burago, Gromov, and Perel'man (see http://seven.ihes.fr/~ …
Manor Mendel's user avatar