"Softness" vs "rigidity" in Geometry

Question

According to common wisdom, there are structures in Geometry that have a more "topological" flavor, others that are more "geometrical", and others that are halfway between. Usually, geometries${}^*$ that are closer to the topological end of the (possibly non totally ordered) spectrum are referred to as being "soft", whereas geometries closer to the geometric end are considered "rigid". For example, topological, differentiable and symplectic manifolds are considered soft, while Riemannian and complex analytic manifolds are considered rigid. And algebraic varieties are even more rigid geometric objects than analytic manifolds.

I've noted there's also a tendency in deformation theory to use the term "rigid" in the opposite way: rigid is something you cannot deform ${}^{**}$, so e.g. compact differentiable manifolds are rigid in this sense (thanks to a well known theorem of Ehresmann), whereas usually algebraic varieties admit nontrivial deformations so they are usually not "rigid".

Also, near the ends of the spectrum (homotopy theory, geometric topology; finite geometries, arithmetic geometry) a more combinatorial/discrete/algebraic approach seems to prevail, while in the middle of the spectrum (metric spaces, Riemannian geometry, analytic and complex algebraic geometry) there seems to be a more "continuous" character.

How to make the above distinctions topology vs geometry and/or softness vs rigidity formally more rigorous?

What are other features of the softness vs rigidity phenomenon?

Let's start with some loose ideas about what I have the impression to be features typical of topology vs geometry.

1. Local invariants. Is the structure at a point distinguishable from the other points or from points of other spaces, or do they all locally look the same? In the case of topological, differentiable and symplectic manifolds there are no local invariants. This also happens for complex analytic manifolds, that we most often regard as instances of a rigid geometry, so it is certainly not a sufficient characterization. For Riemannian manifolds the curvature is a nontrivial local invariant (even a punctual one, in the sense that, even if its definition involves a neighbourhood, differences can be checked pointwise). In algebraic geometry we must be more precise as for the meaning of "locally": locally (w.r.t. some Grothendieck topology) or infinitesimally locally or formally locally? All smooth varieties over a field formally locally look like affine space, but look different locally in the Zariski topology. A condition often put on principal bundles is local isotriviality (i.e. local triviality in the étale topology); this subtlety doesn't appear for vector bundles (aka locally free sheaves).

2. Robustness vs deformability If you perturb the structure in some way, the resulting structure stays isomorphic. The perturbation can be a deformation in the sense of deformation theory, or just picking a close enough datum (e.g. a Riemannian metric close to the original one in the $\mathcal{C}^k$ topology). By Ehresmann, compact differentiable manifolds are invariant under deformations; but noncompact differentiable manifolds are not, indeed starting from dimension $4$ there may be fibrations with homeomorphic non diffeomorphic fibers, so in some sense it's a less purely topological feature. In analytic and algebraic geometry, projective spaces are invariant under deformation; and I have the impression that it also happens for combinatorially/algebraically defined varieties (e.g. toric).

3. Discreteness of moduli. This is somehow the global version of the previous point. A "deformation invariant" structure need not have trivial moduli, but while topological structures tend to have discrete moduli spaces (in whatever sense we intend "moduli space"), when moduli are nondiscrete it means there is something geometric going on. For example, if I remember correctly, line bundles on toric varieties have discrete moduli (the Jacobian is trivial), maybe because they depend only on the combinatorics of the orbits, which is -with some stretch of the meaning- a topological thing.

4. Homogeneity. In many topological categories the "generic" object tend to have a transitive group of isomorphisms (even $n$-transitive sometimes). This happens for topological, differentiable, and (differentiable or real analytic) symplectic manifolds. Of course transitivity of automorphisms also happens for homogeneous spaces in categories of rather "rigid" objects (Riemannian, analytic, algebraic homogeneous spaces), but they are very special objects, not a "random" representative of their category. Anyway they share with more topological categories the possibility of being described combinatorially/algebraically (Schubert cells of the Grassmannian, Lie algebras).

5. Obstructions. In some soft categories we have partitions of unity, which often allow us to patch local data together to obtain a globally defined thing from locally defined things; in more rigid categories this doesn't hold. Also, in rigid categories there are extension problems.

I would say, when two categories of geometric objects are to be compared in softness/rigidity, we are in the following situation. We have categories $\mathcal{C}$, $\mathcal{C}'$ concrete over some base category $\mathcal{S}$ (the latter can be sets, or topological spaces, or any fancy thing like presheaves of simplicial sets over a site). There is a "forgetful functor" $\mathbb{U} :\mathcal{C}\to\mathcal{C}'$ which commutes with the concretizations. We can say (mind that I'm only giving a vague suggestion and I have no idea of a more precise answer) that $\mathcal{C}'$ is obtained by "putting some geometric structure" on objects of $\mathcal{C}$ if one or more of the following holds:

1. $\mathbb{U}$ can make local invariants disappear (e.g. forgetting a Riemannian metric on a manifold).
2. $\mathbb{U}$ can turn a (nontrivially) deformable object into a deformation-rigid one.
3. If there is some notion of moduli spaces $\mathcal{M},\mathcal{M}'$ for objects of $\mathcal{C},\mathcal{C}'$, then the map $\mathcal{M}\to\mathcal{M}'$ induced by $\mathbb{U}$ has nondiscrete fibers.
4. $\mathbb{U}(\mathrm{Aut}(X))\subset\mathrm{Aut}(\mathbb{U}X)$.

${}^*$ (Unfortunately, the term "geometry" is, ambiguously, used both in the general sense of "pertaining to spaces of any kind" and in the more localized sense of "pertaining to structures that are metric or anyway more rigid than topological ones". E.g. the term "differential geometry" is sometimes used as a general header including differential topology, sometimes as a synonym of Riemannian geometry. I hope in the question the ambiguities will be cleared by the context)

${}^{**}$ (Rather than "rigidity", I think a more appropriate term would be "elasticity" or "flexibility", since after all one can fit a rigid variety as special fiber of a family, it's just that the deformed neighbours -i.e. the other fibers- turn out to be isomorphic to the variety you started with)

Answer 1

Take a look at a notion of rigid geometric structure defined by Gromov, see for instance here for the detailed definition and examples. Reproducing the definition here would take a bit too much room, but, I think, it matches the notion of rigidity you have in mind.

Answer 2

Dick Lashof once told me that "differential topology is about first derivatives; differential geometry is about second derivatives".

Answer 3

Here is very rigorous way to think about it.

Doing geometry is like a walking on a tightrope --- you can make few steps and then you fall into algebra (right side?) or into topology (left side?); in any case, you have to start all over again.