On Manifolds: TOP, PDIFF, DIFF, PL

Question

I wonder why in geometric topology, only the following structure of manifolds are emphasized:

• (TOP) topological manifolds
• (PDIFF), for piecewise differentiable
• (PL) piecewise-smooth manifolds
• (DIFF) the smooth manifolds

Can we have more than the ones shown in the graph here: https://en.wikipedia.org/wiki/PDIFF with more subtle or refined structures?

For example, can we consider the affine structure, or Riemannian structure, etc? Or somewhere in between in the straight lines of the figure, can we have other refinement? other structures?

Answer 1

One quick way to get a lot of different structures on manifolds is by considering the transition maps given by local trivializations $f_\alpha:U_\alpha \to \mathbb R^n$.

The procedure is by considering functions $f_{\alpha}f_{\beta}^{-1}:\mathbb R^n \to \mathbb R^n$. If you want smooth a smooth structure you require these maps to be diffeomorphisms, if you want a complex structure you require them to be holomorphic, PL etc.

Answer 2

Other examples of structures on $n$-manifolds:

• For a Euclidean structure, require your transition maps to be isometries of the Euclidean metric on $\mathbb R^n$.
• For a spherical structure, require your your local trivializations to take values in the $n$-sphere $\mathbb S^n$, and your transition maps to be isometries of $\mathbb S^n$, i.e elements of the orthogonal group $O(n+1)$.

In that last example, you can see that it is not necessary for the local trivializations to take values in $\mathbb R^n$ itself, it is sufficient that they take values in some $n$-dimensional manifold.

• For a hyperbolic structure, require your transition maps to be in hyperbolic $n$-space $\mathbb H^n$, and your transition maps to be isometries of $\mathbb H^n$.

In each of these three examples, restricting the local trivializations and the transition maps is used to define a certain kind of geometric manifold. The "model geometry" is a specific Riemannian manifold, and the "transition maps" are elements of the isometry group of that manifold. With that idea in mind, you can start making up your own examples of geometric manifolds.