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In most of the books I read by Tu or Lee, manifolds are always assumed to be smooth (i.e, equipped with smooth structures containing $C^\infty$ charts). Then they move on to define Riemannian manifolds as smooth manifolds with additional structures: tangent spaces over points and a Riemannian metric. That is, at each point $\rho$, the tangent space over $\rho$ is equipped with an inner product $\langle \cdot, \cdot \rangle_\rho $ (positive definite, symmetric) and such that the map $ \rho \mapsto \langle X_p, Y_p\rangle_\rho$ is smooth whenever $X,Y$ are smooth vector fields.

In this question, I'm looking for a notion of $C^k$-Riemannian manifold ($k$ can be 1,2,3,... or even 0?), does such a notion exists. For example, I imagine something like:

  1. A $C^0$-Riemannian manifold is a $C^0$-manifold with additional structures like above but with the map $ \rho \mapsto \langle X_p, Y_p\rangle_\rho$ is only continuous.

  2. A $C^1$-Riemannian manifold is a $C^1$-manifold with additional structures like above but with the map $ \rho \mapsto \langle X_p, Y_p\rangle_\rho$ is $C^1$.

... etc.

If possible, I would like some reference to read on this notions as well. Thank you so much!

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    Your item 1 makes no sense. One can talk about continuous Riemannian metric on a $C^1$-smooth manifold: one needs at least some smoothness to define tangent vectors. The rest are reasonable but you will not be getting geodesics, curvature, etc, unless you assume $C^2$. – Moishe Kohan Jan 21 '24 at 04:06
  • @MoisheKohan, thank you for your answer. From your comment, is it safe to say $C^1$-smooth manifold, equipped with a continuous Riemannian metric in the sense of item 1, is a (maybe) $C^1$-Riemannian manifold? –  Jan 21 '24 at 04:15
  • I would call it a $C^0$-Riemannian manifold. But the concept is of only limited use. Take a look at my answer here regarding natural appearance of low regularity Riemannian metrics. – Moishe Kohan Jan 21 '24 at 04:21
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    BTW, you are probably unaware of this, but every $C^1$-smooth structure has a $C^\infty$ subatlas. Thus, one usually assumes that the manifold itself is $C^\infty$. – Moishe Kohan Jan 21 '24 at 04:27
  • @MoisheKohan Thank you so much, your comments help a lot. And yes, I didn't know that (I'm just an amateur in geometry...). I read your answer in the link, but unfortunately I don't have enough knowledge to understand half of it... I have a small question: What are tangent vectors you mentioned above? I thought a tangent space over a point consists of "tangent vectors"...? –  Jan 21 '24 at 05:04
  • One first defines tangent vectors and then tangent spaces; one does not say a "tangent space over a point" but rather "the tangent space at a point." There are several ways to define tangent vectors for abstract manifolds, you can find this discussed in textbooks; my favorite is via derivations, but one has to be careful with the degree of smoothness (see my answer here). – Moishe Kohan Jan 21 '24 at 14:59

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