Assuming O(D,D) covariant fields as the `fundamental' variables,
Double Field Theory can accommodate novel geometries where a Riemannian metric
cannot be defined, even locally. Here we present a complete classification of
such non-Riemannian spacetimes in terms of two non-negative integers,
(n,nˉ), 0≤n+nˉ≤D. Upon these backgrounds, strings become
chiral and anti-chiral over n and nˉ directions respectively, while
particles and strings are frozen over the n+nˉ directions. In
particular, we identify (0,0) as Riemannian manifolds, (1,0) as
non-relativistic spacetime, (1,1) as Gomis-Ooguri non-relativistic string,
(D−1,0) as ultra-relativistic Carroll geometry, and (D,0) as Siegel's
chiral string. Combined with a covariant Kaluza-Klein ansatz which we further
spell, (0,1) leads to Newton-Cartan gravity. Alternative to the conventional
string compactifications on small manifolds, non-Riemannian spacetime such as
D=10, (3,3) may open a new scheme of the dimensional reduction from ten to
four.Comment: 1+41 pages; v2) Refs added; v3) Published version; v4) Sign error in
(2.51) correcte