All sources on non-Euclidean geometry that I've found so far only consider geometries in which curvature is constant and homogenous in all directions. But what about non-homogenous, hybrid geometries? By that I mean, things like, for example, a 3-dimensional space in which the first dimension is Euclidean, the second dimension is elliptical, and the 3rd dimension is hyperbolic. Is such a thing possible? One could think of such a space as a 3-manifold in the shape of a cylindrical hyperbolic tube, i.e., cross-sections perpendicular to the hyperbolic dimension (of the manifold) are cylinders, while cross-sections parallel to the hyperbolic dimension are hyperboloids of various kinds.

An interesting feature of such geometries, if they are in fact self-consistent, is that orientation matters, and figures in certain orientations may not be able to fit into the space in certain other orientations. For example, a "straight line" parallel to the elliptical dimension is topologically a circle, and you can't rotate it into the Euclidean or hyperbolic dimensions without fundamentally altering its nature. Smaller objects may be rotatable between the inequivalent dimensions, but will experience deformation. Such spaces will also exhibit "preferred" directions, and possibly an absolute reference frame (or at least, an absolute orientation).

An even more interesting question is what happens in higher dimensional hybrid geometries. If we restrict ourselves to only hyperbolic, elliptical, and Euclidean as the only options, then we have the peculiar case that all higher-dimensional hybrid geometries can be partitioned into 3 homogenous subspaces (i.e., a homogenously hyperbolic subspace, a homogenously elliptical subspace, and a homogenously Euclidean subspace, with the sum of their dimensions equal to the dimensionality of the entire space). This is strange because no matter which dimension is chosen to be hyperbolic/elliptical/flat this decomposition is always possible. Objects rotating within each subspace will experience no deformation.

However, if the relative curvatures of, say, the hyperbolic dimensions differ across dimensions, then multiple {hyperbolic,elliptical} dimensions no longer are freely interchangeable, and this ternary subspace decomposition is no longer possible. Rotations between two elliptical dimensions of unequal curvature will experience deformation (and probably resistance due to said deformation -- a kind of deformative resistance?).

Even more interesting things arise if we consider what kind of shapes may exist in such hybrid geometries. We might get a polytope (perhaps even a regular one??!) that's closed along one dimension but unbounded along another dimension, like toroidal, cylindrical, or other such strange things.

Of course, once we step out of the confines of "traditional" non-Euclidean geometry (i.e., homogenous geometries), there's no telling what else we can hybridize geometries with. What about a 5D space produced by the product of an Euclidean line, a circle, a hyperbola, and a projective plane? It could contain polytopes that are part-Euclidean, part bounded, part hyperbolic, and part projective? What manner of strange shapes would be possible in such a hybrid geometry?