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All I meant is this: a regular surface is a certain kind of subset of
Euclidean space while a smooth manifold is a certain kind of topological space endowed with a certain kind of coordinate atlas. They're just different types of objects (albeit with strong connections between them, most notably that
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I think there's an error where the article claims that the surfaces of revolution obtained by revolving e^t or cosh(t) or sinh(t) have constant gaussian curvature -1. This would contradict
Hilbert's theorem of no complete -1 curvature surfaces in E^3. The surfaces obtained are negatively curved, but
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Early in this article, section 'Curvature of surfaces in E^3', the definition of mean curvatures K.sub.m = (ET + GR -2FS) / (1+P^2+Q^2)^2 utilizes quantities not defined up to that point. I believe that (E,F,G) are the parameters of the first fundamental form introduced later in section "Line and
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I don't want to tamper with the article, but would the latest editor of this section or some other dispassionate soul kindly replace E,F,G with 1,0,1 (special case of the introductory discussion) or else define the quantities before they're used? / bruce_bush_nj /
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That sounds like a line I probably wrote here a few years ago. It's a bit clunky and I have no problem with it being changed, but as far as I know "smooth surface" is not considered formal language (except perhaps in algebraic geometry, where the meaning would be
336:(By the way, "regular surface" is a word I've only encountered in curves & surfaces textbooks and never 'in the wild' – although I can see it's out there. My own language would be "smoothly embedded submanifold" or "smoothly embedded surface.")
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Taking a coordinate change from normal coordinates at p to normal coordinates at a nearby point q, yields the Sturm-Liouville equation satisfied by H(r,θ) = G(r,θ)½, discovered by Gauss and later generalised by
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It is intuitively clear that a sphere is smooth, while a cone or a pyramid, due to their vertex or edges, are not. The notion of a "regular surface" is a formalization of the notion of a smooth surface.
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is not clear. What is the link between a normal coordinate changes at p to q, and the equation Hrr = – K H. ? Why Hr is the
Jacobian of this coordinate change ? Thank you for your explanations.
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the second sentence could be changed to something like "A 'regular surface' is a formal notion of surface which excludes such non-smooth phenomena." (But this is still rather clunky.)
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Interesting. I've also never seen "regular surface" 'in the wild', but in my world the word "smooth surface" is used frequently to refer to smooth 2-manifolds.
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Made some edits to fix that. Not sure that that section is even needed, there is much better discussion of surfaces of constant curvature later in the article.
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because every point in every surface has at least one straight line running through it, for example, the line coinciding with the normal vector.
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Sure, I agree. But 'smooth 2-manifold' is not synonymous with 'regular surface'. (A regular surface is a smooth 2-manifold but not vice versa.)
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This classical computation is discussed for example in Berger's book. I'll give you a detailed explanation myslef, if I have time.
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Ruled surfaces are surfaces that have, through each point, at least one straight line lying entirely within the surface.
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I find this troubling because far as I know, "regular surface" and "smooth surface" are synonyms.
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The notion of a "regular surface" is a formalization of the notion of a smooth surface.
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Definition of mean curvature utilizes undefined quantities E,F,G introduced later ?
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I was under the impression that they are. Do you know a counterexample?
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area elements", or possibly the (e,f,g) of the second fundamental form.
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have at least one straight line running through every point"": -->
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Ah yes, of course! I was being very silly. I see what you mean now.
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a regular surface possesses a canonical smooth manifold structure).
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