Examples of finitely differentiable manifolds

Question

I am trying to get a more intuitive grasp of the concept of a manifold. In physics, we are usually interested in smooth manifolds, i.e. infinitely differentiable ones. In the literature, however, we are often assuming a $C^k$ differentiable manifold, where $k$ could be some positive integer. What are some simple examples of finitely differentiable manifolds, such as for $k=0,$ $1,$ or $2$, say?

Answer 1

Well, you can take any smooth manifold, and just consider it as a $C^k$ manifold for any $k<\infty$. (If you define a manifold in terms of a maximal atlas, this means you enlarge the atlas to contain all charts that are $C^k$-compatible with your smooth atlas, instead of just smoothly compatible charts.) This is the only way to get an example for $k>0$: it is a (nontrivial) theorem that every $C^k$ manifold comes from a $C^\infty$ manifold (which is unique up to $C^\infty$ diffeomorphism) in this way. For $k=0$, there exist $C^0$ (i.e., topological) manifolds that do not come from smooth manifolds in any dimension greater than $3$, but they are quite difficult to construct.

That said, there are certainly easy examples of $C^k$ manifolds which are not $C^\infty$ manifolds in any particularly "natural" way. For instance, let $U\subseteq \mathbb{R}^m$ and take the image of an embedding $f:U\to\mathbb{R}^n$ which is $C^k$ but not $C^{k+1}$. Very concretely, in the case $m=1$, for instance, you can just take a curve defined by $n$ real-valued functions on an interval such that there is no point where their derivatives all vanish (so you get an embedding) but one of the functions is only $C^k$. Of course the image of such an embedding can be given a smooth structure by identifying it with $U$, but typically you will be interested in it as a submanifold of $\mathbb{R}^n$ and it is only $C^k$ as a submanifold.

Answer 2

The graph of a $C^k$ function $f\colon \mathbb R\to\mathbb R$ is an example of a $C^k$ manifold. For instance, consider the graphs of the functions $|x|\cdot x^k$.