Taking $mathbf{O}(D,D)$ covariant field variables as its truly fundamental constituents, Double Field Theory can accommodate not only conventional supergravity but also non-Riemannian gravities that may be classified by two non-negative integers, $(n,bar{n})$. Such non-Riemannian backgrounds render a propagating string chiral and anti-chiral over $n$ and $bar{n}$ dimensions respectively. Examples include, but are not limited to, Newton--Cartan, Carroll, or Gomis--Ooguri. Here we analyze the variational principle with care for a generic $(n,bar{n})$ non-Riemannian sector. We recognize a nontrivial subtlety for ${nbar{n} eq 0}$ that infinitesimal variations generically include those which change $(n,bar{n})$. This seems to suggest that the various non-Riemannian gravities should better be identified as different solution sectors of Double Field Theory rather than viewed as independent theories. Separate verification of our results as string worldsheet beta-functions may enlarge the scope of the string landscape far beyond Riemann.
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