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Hilbert space?[edit]

I am new to topology. It appears that a Riemannian manifold is a manifold in which the tangent space is a Hilbert space. Is that all there is to it? —Ben FrantzDale 01:15, 5 May 2007 (UTC)[reply]

A finite-dimensional one over the reals, but yes. Also the inner product must vary smoothly from one tangent space to the next. -- Fropuff 16:41, 5 May 2007 (UTC)[reply]
Thanks. Why does it have to be finite-dimensional? —Ben FrantzDale 17:14, 5 May 2007 (UTC)[reply]
It has to be the same dimension as the manifold and manifolds are usually finite dimensional. —Preceding unsigned comment added by (talk) 18:52, 18 March 2010 (UTC)[reply]

Riemannian metrics – a counter example[edit]

I think that it would be useful to have a simple example in which the inner product is not continuous. TomyDuby (talk) 18:12, 7 September 2008 (UTC)[reply]

Would this be better delt with here or at metric tensor? It seems more relevant to understanding the tensor itself rather than understanding what a Riemannian manifold is. An example is, at any rate, easy to give. On Euclidean space Rn, define
-siℓℓy rabbit (talk) 20:15, 7 September 2008 (UTC)[reply]
Thanks for the count erexample. Again, I learnt something!
Where to put it? I think that it might be better at metric tensor. TomyDuby (talk) 20:37, 11 September 2008 (UTC)[reply]

Notation S^2[edit]

I'm in a grad-level geometry course right now, and neither in the course nor any of the texts I've used have I seen the notation that occurs here "More formally, a Riemannian metric is a section of the vector bundle ." What is meant by here? Is this like a subbundle operator on ? Can someone who knows either describe the notation (in the article). I think the intention is that is the subbundle of the tensor bundle that is the disjoint union of 2-tensor fields , where the subbundle selects those tensor fields that are symmetric and positive definite.

The notation I have seen for this would be 'a Riemannian metric is a smooth section of ' (using Lee's notation, or maybe you could say it's a smooth section of , though I don't know if that notation is used anywhere), such that the tensor field is symmetric and positive definite. --Tekhnofiend (talk) 00:26, 28 February 2011 (UTC)[reply]

I interpret here as being the symmetric square of the cotangent bundle. I may try to add this clarification. --Dylan Moreland (talk) 20:12, 7 May 2011 (UTC)[reply]

Possible plagiarism?[edit]

Hi, I've noticed that in the section "Riemannian metrics" the "Examples" subsection is taken word for word from Do Carmo's book, is this a problem? (talk) 03:54, 18 January 2013 (UTC)[reply]

Yes, it's definitely a problem. The content in question was added in this pair of edits that substantially expanded the article. This is going to need to be examined carefully to determine if there is more infringing content. Sławomir Biały (talk) 23:01, 19 January 2013 (UTC)[reply]
This is bad... (just saying). -- Taku (talk) 04:21, 20 January 2013 (UTC)[reply]

Is that all it is?[edit]

A Riemann space is defined in the lede as "a real smooth manifold M equipped with an inner product on the tangent space at each point that varies smoothly".

Doesn't that also describe an ordinary sphere in R³ (Euclidian 3-Space)? -- (talk) 04:24, 20 May 2013 (UTC)[reply]

Well, indeed, the "ordinary" sphere (seen as a submanifold of R3, inheriting by restriction the usual inner-product) is an example of a Riemannian manifold. See the last paragraph (starting with "Every smooth submanifold") of section "Overview". In general, however, there is no need to immerse a manifold in some Euclidean space in order for it to acquire a metric; normally, you would define a manifold regardless of an immersion (i.e., without reference to an ambient Euclidean space), and you would simply add the metric (as some extra structure), as required by the context; that is basically what this whole article is about. Pierdeux (talk) 17:37, 12 June 2013 (UTC)[reply]

Introduction: mention differentiability of vector fields?[edit]

Should the introduction mention that the vector fields X and Y have to be differentiable (otherwise there probably are choices for X and Y that would make fulfilling the smoothness condition of g impossible), or is this clear enough from context? (talk) 11:32, 24 May 2013 (UTC)[reply]

I have just independently noticed (the differentiability of the fields X and Y is stated now) that the constraint of differentiability of X and Y is insufficient. For every nondegenerate g, one can always choose differentiable X and Y such that g(X, Y) is non-smooth. This implies that no manifolds meet this definition. Some repair is evidently needed. —Quondum 14:01, 29 January 2020 (UTC)[reply]
Smoothness is totally irrelevant to the actual definition of a Riemannian metric, especially when it comes to much modern work. Shouldn't the introduction just say that one assigns a positive-definite inner product to each tangent space?
Gumshoe2 (talk) 03:38, 3 May 2020 (UTC)[reply]
If one wants to define smoothness of the metric, the best way to do it (since it trivially extends to defining any desired regularity of the metric) is to use local coordinates. I'll make the edits
Gumshoe2 (talk) 03:41, 3 May 2020 (UTC)[reply]

Incorrect classification of article?[edit]

I don't know if this is the right place to state these concerns, but I think that there are three significant issues about the way this article is classified and written (feel free to disagree with me):

Firstly, shouldn't this article be characterized under WikiProject Geometry?

Secondly, Riemannian manifolds are one of the most principal and fundamental objects of study in geometry (particularly in differential geometry), so shouldn't this page be rated as 'High Importance' at the least? Moreover, the article about the metric tensor, which is fundamentally related to this article, is rated high-importance, so should this not be as well? (Perhaps the two should be merged?)

Thirdly, because of the importance of Riemannian manifolds in geometry, I think that the intro needs a touch-up (in particular it should contain a non-technical piece for the layman; I may put this in when I get time). SillyBunnies (talk) 06:07, 22 June 2014 (UTC)[reply]

Differential properties enabled by a metric tensor[edit]

I agree in with this edit of the lead. On a differentiable manifold without a metric tensor, a form of gradient of a scalar field (the index-lowered version, which is a co-vector field) is still well-defined, and the index-raised version is, in itself, not adding enough to merit a mention here, especially considering that it is not always clear whether "gradient" refers to the index-lowered or -raised version and this would need clarification. A similar (but trickier) argument applies to the divergence of a vector field: the n-form derivative of (n – 1)-form (e.g. the current density 3-form Jαβγ) is well-defined and is similar to a scalar; this is related to the divergence of a vector field using considerably less additional structure than a metric tensor. The subtleties and lack of evident added value brought by the metric tensor here make their inclusion in the lead unsuitable.

However, there is differential structure that is enabled by the metric tensor. The above two operations cannot be combined without it, and in particular the full structure of a metric tensor is necessary (AFAICT: this is WP:OR) for any form of the Laplace operator to be well-defined. Wave equations (or harmonic equations in the Riemannian context) are consequently only meaningful in the presence of a metric tensor. This seems like a worthwhile addition to the article, if it can be sourced. —Quondum 13:31, 14 August 2018 (UTC)[reply]

Yes, defining the Laplace operator would require the metric. JRSpriggs (talk) 23:54, 14 August 2018 (UTC)[reply]

The induced metric[edit]

I think that all the bullet points starting from the second one in the Examples section are actually about different ways that a metric can be induced and how to carry that process mathematically... I believe they would rather belong in a Immersed manifold/induced metric section, each in different subsections. — Preceding unsigned comment added by Vyrkk (talkcontribs) 21:53, 27 October 2019 (UTC)[reply]


I reorganized the page, added some things, and deleted others. I think it reads better now. I plan to add a section on curvature and the geodesic equation. I've left the very last section exactly as is since it doesn't naturally go into the other one, but it'll be subsumed into the eventual part on the geodesic equation.

Sorry if I changed too much or deleted something I shouldn't have. It seemed like some of the material was transplanted from books that obscured some of the point, and there was a lot of repetition and unclear statements. I tried to make sure that what I added is on a readable level.

Gumshoe2 (talk) 09:00, 3 May 2020 (UTC)[reply]