Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
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).
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).
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.
Homogeneity. In many topological categories the "generic" object tend to have a transitive group of isomorphisms (even $n$-transitive sometimessometimes). 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).
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.
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).
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).
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.
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).
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.
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).
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).
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.
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).
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.
