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on a smooth manifold by defining an inner product on each tangent space, pointwise, in a smooth way. to pass from a notion of length of a tangent vector to length of a curve on the manifold, all you have to do is integrate. There is a lot more you can do with
Riemannian geometry, but if you can grasp that starting notion, then the rest should fall into place. I guess what I'm saying is, if you think the article is incomprehensible, you may be right. Asking for clarification might be a good idea....
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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
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I added a paragraph to the article. I should probably look it over again tomorrow to make sure it's coherent, but I want to make the basic idea of a
Riemannian manifold transparent; an inner product gives you a notion of length on a vector space, and a Riemannian metric gives you a notion of length
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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
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Hi Lethe, relatively new contributor here. What I'm thinking would be useful, is some initial mention for motivation -- why did
Riemann introduce this concept, what could it effectively model? Including a few sentences outlining such information, optimally as a continuance of the first paragraph,
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You know, the article is much too short. It could stand to have many examples and explanatory sentences added. Perhaps you could help us decide how best to improve it, with focused questions. at what point does the exposition lose you? if you ask for more, you might receive it... -
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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?
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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.
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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
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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
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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).
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I agree with the idea that this article was too small and not enough documented. That's why I have added some further things regarding the metric and the metric space ... Hope this is coherent for everybody
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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
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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):
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You were right to delete the link, thanks. I found two more articles with the same link and also deleted it there. However, for future reference, you accidentally deleted a bit more than just the link:
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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
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Is there any chance that one could have a paragraph (or two) about What this means with less of the math? I realize that one would have to make assupmtions etc. in order to explain this to others.
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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
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I went back and added a few section headers. Lumping everything under
Introduction seems inappropriate. Actually, a more thorough list of properties should probably be added.
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I know from classical differential geometry that one needs the vector product in order to calculate the metric tensor. But these two things are surely not one and the same.
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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".
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of a
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from the space of bilinear mappings on the tangent bundle to the tensor product of the cotangent bundle with itself. More details can be found in the
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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?
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I have not yet given up the hope to understand this article. Let me start with the beginning: what confuses me in the first sentence is that
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that substantially expanded the article. This is going to need to be examined carefully to determine if there is more infringing content.
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154:-- I think my new paragraph here kinda sticks out like a sore thumb. I don't like it, and perhaps it belongs in a different article (
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A finite-dimensional one over the reals, but yes. Also the inner product must vary smoothly from one tangent space to the next. --
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Secondly, Riemannian manifolds are one of the most principal and fundamental objects of study in geometry (particularly in
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the metric (as some extra structure), as required by the context; that is basically what this whole article is about.
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900:{\displaystyle g_{p}(v,w)={\begin{cases}v_{p}\cdot w_{p}&p\not =0\\2v_{p}\cdot w_{p}&p=0\end{cases}}}
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Riemann for Anti-Dummies. The site was of questionable authority and had a very non-neutral point of view.
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is used for two different purposes: once it is the metric tensor and second time it is the inner product.
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I am new to topology. It appears that a Riemannian manifold is a manifold in which the tangent space is a
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I think that it would be useful to have a simple example in which the inner product is not continuous.
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It has to be the same dimension as the manifold and manifolds are usually finite dimensional.
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I have made some changes to the offending section. Let me know if it is clearer. Thanks,
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The notation I have seen for this would be 'a Riemannian metric is a smooth section of
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is the subbundle of the tensor bundle that is the disjoint union of 2-tensor fields
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the metric tensor. They are one and the same thing, by the canonical isomorphism
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If you wish to start a new discussion or revive an old one, please do so on the
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Firstly, shouldn't this article be characterized under WikiProject Geometry?
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571:{\displaystyle \{\left({\frac {\partial }{\partial x_{i}}}\right)_{p}\}_{i}}
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Doesn't that also describe an ordinary sphere in R³ (Euclidian 3-Space)? --
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I have just independently noticed (the differentiability of the fields
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Yes, it's definitely a problem. The content in question was added in
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I've come to terms with the fact that I will never understand this.
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would substantially improve the overall quality of this article.
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a manifold in some Euclidean space in order for it to acquire a
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of the cotangent bundle. I may try to add this clarification. --
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Well, indeed, the "ordinary" sphere (seen as a submanifold of
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Yes, defining the Laplace operator would require the metric.
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is stated now) that the constraint of differentiability of
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Thanks for the count erexample. Again, I learnt something!
99:. Please leave them next time. Anyway, I fixed it now. --
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Introduction: mention differentiability of vector fields?
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Where to put it? I think that it might be better at
1240:{\displaystyle TM\times TM\rightarrow \mathbb {R} }
993:Thanks. Why does it have to be finite-dimensional?
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306:{\displaystyle L^{2}(TM)\to T^{*}M\otimes T^{*}M}
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473:{\displaystyle {\mathcal {V}}(M)}
412:{\displaystyle {\mathcal {V}}(M)}
206:Question about the first sentence
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1379:{\displaystyle S^{2}(T^{*}\!M)}
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1273:{\displaystyle \Lambda ^{2}M}
1103:{\displaystyle S^{2}T^{*}M\,}
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657:) giving (0, ..., 1, ..., 0)?
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480:. Thanks for clearing this.
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1316:{\displaystyle (T^{*})^{2}M}
363:Riemannian metrics: Question
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1200:{\displaystyle S^{2}T^{*}M}
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632:Is the derivative ∂ / ∂
70:Riemann for Anti-Dummies
18:Talk:Riemannian manifold
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1067:vector bundle
1064:
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1018:82.31.208.151
1015:
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986:
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968:Hilbert space
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942:metric tensor
939:
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764:metric tensor
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319:metric tensor
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94:
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28:
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19:
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1698:
1694:
1690:
1672:
1645:— Preceding
1641:
1601:
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1590:
1583:
1576:
1566:
1550:SillyBunnies
1546:
1535:
1532:
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1506:
1502:
1498:
1494:
1490:
1473:
1470:
1409:
1338:I interpret
1249:
1040:
1037:Notation S^2
965:
914:siℓℓy rabbit
767:
746:
692:siℓℓy rabbit
651:
644:
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616:
612:
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447:Yes I meant
426:siℓℓy rabbit
420:
368:
366:
324:siℓℓy rabbit
237:
233:
216:
211:
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132:
118:
101:Jitse Niesen
73:
60:
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1325:Tekhnofiend
1012:—Preceding
36:This is an
1717:such that
367:What is ν(
85:categories
1624:JRSpriggs
1570:this edit
770:, define
321:article.
61:Archive 1
1771:Gumshoe2
1747:Gumshoe2
1659:contribs
1647:unsigned
1511:Pierdeux
1014:unsigned
946:TomyDuby
749:TomyDuby
723:TomyDuby
669:TomyDuby
482:TomyDuby
373:TomyDuby
347:TomyDuby
345:Thanks.
219:TomyDuby
121:E.Lefraw
1731:Quondum
1614:Quondum
1499:immerse
1495:example
1065:of the
1063:section
982:Fropuff
650:, ...,
192:Zadigus
180:Geo.per
169:Geo.per
39:archive
1743:space?
1503:metric
1651:Vyrkk
1605:WP:OR
1061:is a
232:Yes,
160:Lethe
149:Lethe
140:Lethe
95:like
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1713:and
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1680:talk
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1554:talk
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1022:talk
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753:talk
727:talk
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511:Let
486:talk
433:talk
377:talk
351:talk
331:talk
223:talk
196:talk
105:talk
91:and
1595:αβγ
1507:add
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615:in
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1296:∗
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1230:→
1221:×
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944:.
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589:∈
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288:⊗
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882:p
875:p
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862:p
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847:0
844:≠
841:p
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830:w
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89:]
50:.
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