Definition talk:Riemannian Metric
Why $C^\infty$
>If you are asking for motivation, then I guess this is simply the best behaved setting. For tangent spaces C^1 is ok, but here it is implicitly implied that you should be able to study structures differentiable to arbitrary orders.
I think it is sufficient to assume $C^1$ for this definition in order to be able to study structures differentiable to arbitrary orders, because $C^k$ cases are included in $C^1$ cases.
On the other hand, if we assume here $C^\infty$, then we cannot discuss $C^k$ cases anymore.
Yes, some author prefers to assume $C^\infty$ everywhere in his book for simplicity, but we also need a more general definition in $\mathsf{Pr} \infty \mathsf{fWiki}$. Having a version for $C^1$ and a version for $C^\infty$, separately, thought they are the same thing, is stupid.
This is a common problem here. --Usagiop (talk) 22:52, 21 September 2023 (UTC)
- What, that the contributors are stupid? --prime mover (talk) 05:13, 22 September 2023 (UTC)
- Okay then, your mission (you've seen it being done, it's trivially easy to follow) is to restructure this page with an "also defined as" and document the stupidity in that (perhaps even putting the documented definition into a "let's laugh at the authors' stupidity" Mistakes section (according to the standard $\mathsf{Pr} \infty \mathsf{fWiki}$ way of working, it's trivially easy to do, you just have to find examples of how this is done and copy the technique), then amend the definition so it is correct, and then make sure that every page which implements this has been checked to make sure that all the dependencies on this page do not depend on the stupid implementation, and use your preferred perfect implementation.
- Soon as possible, please. --prime mover (talk) 05:19, 22 September 2023 (UTC)
- I'm interested to discover what else comes across as suboptimal: "This is a common problem here". We try to document as many approaches as possible. That's the aim. --prime mover (talk) 17:07, 22 September 2023 (UTC)
- When I need a definition, there often already exists one but only for a special case. For example, Definition:Borel Measure is currently only defined for $\R^d$. Why not for general topological spaces? The problem is that a page already exists but for some unessential reason, restricted to a special case. I am forced to do a hard refactoring work to add a slightly more general definition I need. --Usagiop (talk) 02:10, 23 September 2023 (UTC)
We have the definition that has stood the test of time, that is the Riemannian manifold which is defined to be a smooth manifold equipped with a Riemannian metric. Accordingly, a Riemannian metric is a family of smoothly varying inner products on the tangent spaces of a Riemannian manifold (smooth manifold). Finitely differentiable manifolds do exist and they deserve a separate name and treatment, but better to keep Mr. Riemann out of this picture. Unless you can provide with a source, where it is explicitly stated that Riemannian manifold is k-times differentiable manifold etc. --Julius (talk) 07:45, 22 September 2023 (UTC)