Is there some example of such object? I have been looking some articles on quantum spheres, but a lot of families seem to be isospectral and then no questions are asked about limit cases.
I am particularly interested on the simplest (?) case where a 2-sphere is deformed down to one point, and on the one when a spherical 3-manifold is schrinked down to $S^1$. This case could perhaps come standalone, or perhaps with some recipe to consider it as a fiber bundle with a 2-sphere base.
The motivation is to look what happens with the group of isometries over such spaces. For $S^3$ it is SO(4)~SU(2)xSU(2) it goes down to the circle with isometry group U(1), so the whole interpolation could look a lot as the Higgs mechanism in physics, where two parameters, quartic and quadratic self couplings of the higgs field, interpolate between the unbroken group and the reduced one.