Motivations of studying $C^r$ manifolds when $r<\infty$

Question

It seems to me that most differential geometers or topologists only care about smooth manifolds, i.e. the manifolds that are infinitely differentiable. My question is, does the study of $C^r$ manifolds when $r<\infty$ have any motivations from other areas or the real world? Are there any "good" examples that make the study of such manifolds necessary?

Answer 1

I will tend to argue that the study of $C^r$-manifolds with $1\leqslant r<+\infty$ is unnecessary due to to the following result:

Theorem. Let $k<k'\leqslant+\infty$, then a $C^k$-manifold is $C^k$-diffeomorphic to a $C^{k'}$-manifold and if two $C^{k'}$-manifolds are $C^k$-diffeomorphic, they are $C^{k'}$-diffeomorphic.

Proof. See the chapter $2$ of Differential Topology by Morris Hirsch. $\square$

This result means that the classification of $C^k$-manifolds or $C^{\infty}$-manifolds are the same, these notions are indistinguishable from the topological point of view.

Nevertheless, some dynamical problems involving iteration of $C^k$-maps or $C^k$-foliations with $k<+\infty$ are meaningful, see for example the chapter $3$ of Geometrical Methods in the Theory of Ordinary Differential Equations by Vladimir Arnol'd or the book Introduction to the modern theory of dynamical systems by Anatole Katok and Boris Hasselblatt.

Answer 2

To expand on the remark at the end of C. Falcon's answer, I'd like to make some comments from the point of view of dynamical systems. What I write above is nowhere near comprehensive; it's a sampler.

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In the study of dynamical systems that involves some differentiability (e.g. smooth ergodic theory), constructing invariant objects out of dynamics typically costs some regularity, as the acting group typically is not compact, so the constructions involve limits. For instance the so-called stable and unstable foliations for an Anosov diffeomorphism are only guaranteed to be Hölder continuous transversally, even when the diffeomorphism is real analytic (along leaf directions these foliations will be as regular as the diffeomorphism).

The theorem C. Falcon cites translates into the w.l.o.g of taking the ambient manifold to be $C^\infty$; matters of finite regularity comes into play when one is interested in its subobjects (immersed/embedded submanifolds or foliations). More generally, the interesting mathematics typically is not due to the finite differentiability of the charts of the manifold but the spaces of finitely differentiable functions on manifolds (though of course one should keep in mind the works of Milnor, Kervaire, Sullivan etc. as well as the moduli space of smooth structures questions for topological manifolds (possibly carrying some more structure, e.g. some specified dynamics)). Changing the regularity changes the function spaces as well as the topology, which governs the quality or strength of perturbations and genericity as well as the equivalences between different dynamics, involved (consider the fact that $C^r$ is Banach if $r<\infty$ but Fréchet if $r=\infty$). My own speculation is that back when the part of differential topology in discussion was first being formalized category theory was popular too, and $C^r$ manifolds got coupled with $C^r$ maps, diffeomorphisms and vector fields to give the $C^r$ category, which certainly is appropriate from a categorical point of view (see e.g. Lang's Fundamentals Of Differential Geometry). The questions such as regularity of composition of functions are natural questions from this point of view.

Finally let me remark that depending on the type of dynamics the range of regularity that is more relevant changes. For instance for (so-called nonuniformly) hyperbolic dynamics often $C^{1+\text{Hölder}}$ is relevant (whereas $C^1$ does not work under the traditional assumptions, see e.g. Pugh's "The $C^{1+\alpha}$ hypothesis in Pesin Theory). In KAM theory (which one may think of as elliptic dynamics) higher regularity is often required (higher regularity perturbations are more delicate perturbations, see e.g. https://math.stackexchange.com/a/3892256/169085 for the formalism), see e.g. Rodríguez Hertz' "Stable ergodicity of certain linear automorphisms of the torus" (https://annals.math.princeton.edu/2005/162-1/p02) (which is related to partially hyperbolic phenomena as well).