A non-analytic manifold

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Torn Apart By Dingos
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A non-analytic manifold

Postby Torn Apart By Dingos » Sat Dec 27, 2008 10:54 am UTC

I'd like an example of a manifold that isn't analytic (or differentiable). A silly example is the real line with the two charts [imath]\varphi(x)=x, \psi(x)=2x[/imath] if x>0, x otherwise.. Then [imath]\varphi^{-1}\circ \psi[/imath] isn't differentiable at 0. But this can easily be turned into an analytic manifold by simply dropping one of the charts. Is there a manifold that can't be given charts so as to make it analytic (or differentiable)?

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Re: A non-analytic manifold

Postby demon » Sat Dec 27, 2008 12:05 pm UTC

http://mathworld.wolfram.com/SmoothManifold.html
says yes - well, at least it says that there exist non-smooth manifolds in R^4 and higher. I can't give you the example though, I just accidentally stumbled upon this while preparing for a test on (among others) C^1 manifolds.

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Re: A non-analytic manifold

Postby dp » Sat Dec 27, 2008 1:21 pm UTC

My knowledge on this is fairly limited, but my understanding of manifolds were that they were a topological space endowed with an atlas of charts. So, the question you're asking is "is there a topological space that can't be endowed with a differentiable algebraic structure"? I don't know the answer, but I'd imagine it is "yes" and that the resulting space is fairly funky.

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Re: A non-analytic manifold

Postby taby » Sat Dec 27, 2008 3:22 pm UTC

The surface of a quaternion Julia set (a fractal) is non-differentiable.

If you're interested, there is a good paper on this:
Cochran WO, Lewis RR, Hart JC. The normal of a fractal surface. The Visual Computer. Volume 17, Number 4; 2001

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Re: A non-analytic manifold

Postby Torn Apart By Dingos » Sat Dec 27, 2008 3:40 pm UTC

demon wrote:http://mathworld.wolfram.com/SmoothManifold.html
says yes - well, at least it says that there exist non-smooth manifolds in R^4 and higher. I can't give you the example though, I just accidentally stumbled upon this while preparing for a test on (among others) C^1 manifolds.

Thanks. So apparently this was a more complex question than I'd thought.

dp wrote:My knowledge on this is fairly limited, but my understanding of manifolds were that they were a topological space endowed with an atlas of charts. So, the question you're asking is "is there a topological space that can't be endowed with a differentiable algebraic structure"? I don't know the answer, but I'd imagine it is "yes" and that the resulting space is fairly funky.

Exactly right.

taby wrote:The surface of a quaternion Julia set (a fractal) is non-differentiable.

If you're interested, there is a good paper on this:
Cochran WO, Lewis RR, Hart JC. The normal of a fractal surface. The Visual Computer. Volume 17, Number 4; 2001

Is that a non-differentiable manifold though? Any parametrizable non-differentiable curve or surface is an analytic manifold if you take as its atlas the single chart that is its parametrization.

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Re: A non-analytic manifold

Postby taby » Sat Dec 27, 2008 9:42 pm UTC

You can't define an actual dyed-in-the-wool surface normal for the 3D quaternion Julia set (approximation is the best that one can achieve), because there is no limit to the fineness of surface detail (unlike, say, the surface of a solid 3D ball). And so there is no tangent plane, principle curvature planes, differentiation, etc. I've demonstrated an approximation recently via marching cubes, if you are interested: http://cavekitty.ca/inv_ssa.pdf

You were asking for a surface that "isn't analytic (or differentiable)". Sorry if I misread. Perhaps I'm not understanding what you were looking for.
Last edited by taby on Sat Dec 27, 2008 10:06 pm UTC, edited 5 times in total.

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Re: A non-analytic manifold

Postby t0rajir0u » Sat Dec 27, 2008 10:02 pm UTC

You might be interested in Donaldson's theorem.

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Re: A non-analytic manifold

Postby skeptical scientist » Sun Dec 28, 2008 10:32 am UTC

taby wrote:You were asking for a surface that "isn't analytic (or differentiable)". Sorry if I misread. Perhaps I'm not understanding what you were looking for.

His question was, "Is there a manifold that can't be given charts so as to make it analytic (or differentiable)?" Your manifold is not differentiable as a hypersurface in R4, but it may be homeomorphic to a smooth manifold, which means there are charts which make it into a differentiable manifold.
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Re: A non-analytic manifold

Postby taby » Sun Dec 28, 2008 4:15 pm UTC

Is this because the surface area is finite?

Douady and Hubbard showed that these fractal sets (well, the Mandelbrot anyway) are connected, so the appearance of disconnected islands is only an artifact of approximation, so I can see how they could be akin to a solid 3D ball in that they are both in "one piece". I assume this is a necessity? e.g.: You cannot map one sphere to two spheres?

I'm trying to wrap my head around this coming from the perspective of simple texturing in computer graphics. I don't see off-hand how to parameterize the surface absolutely (doing it using an approximated surface is "easy"). Any info is greatly appreciated! Thank you. :)

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Re: A non-analytic manifold

Postby Yakk » Mon Dec 29, 2008 9:13 pm UTC

t0rajir0u wrote:You might be interested in Donaldson's theorem.

Yes.
Wikipedia wrote:On the other hand, there exist topological manifolds which admit no differential structures, see Donaldson's theorem (compare Hilbert's fifth problem).

It also seems that there aren't any in dimensions below 4?

There also seems to be an implied construction via Freedman on the wiki page.
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Re: A non-analytic manifold

Postby MartianInvader » Tue Dec 30, 2008 8:08 pm UTC

It is my understanding that an "analytic" manifold is a different concept than a "differentiable" manifold. A differentiable manifold is one that can be given charts in Euclidean space (R^n) so that the transition maps are smooth. An analytic manifold is one that can be given charts in complex space (C^n), so that the transition maps are complex differentiable, or analytic. All analytic manifolds are differentiable, but not vice versa.

Trivial examples of differentiable manifolds with no complex structure would include anything with odd dimension. Slightly less trivial examples would be even-dimensional real projective space.

Now there's also the distinction between topological manifold (has charts to Euclidean space) and differentiable manifold (has smooth transition maps). As has been mentioned, all topological manifolds of dimension 3 or less can be given a differentiable structure. I've seen the construction in dimension 4 of a topological manifold which isn't smooth, but I don't remember it exactly (it's fairly involved). I think it involved quotienting out by a wild arc, and something to do with cones on CP^2 - sorry, I don't remember exactly how it worked.

It's also worth mentioning that the same topological manifold can be given multiple differentiable structures which are not diffeomorphic. In fact, John Milnor proved several decades ago that the seven-dimensional sphere has 28 distinct differentiable structures. That is, there are 28 different manifolds, none of which are diffeomorphic to each other, but all of which are homeomorphic to the 7-sphere. (Check it out!)

Edit: After checking wikipedia, I realized I was confusing the terms "analytic manifold" with "complex manifold", both of which are distinct from "differentiable manifold." The examples I gave were differentiable manifolds that aren't complex manifolds. I suspect that any differentiable manifold can be made real analytic.
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