Re: [math-fun] Affine manifolds
That Wikipedia article mentions two other conjectures beside the one I cited. The first one refers to affine manifolds "of constant volume" -- a notion I don't understand.* And which as far as I can tell is found nowhere in the reference cited (M. Hirsch & W. Thurston, Annals, 1975) The third conjecture strikes me as highly technical. But a manifold's having Euler characteristic = 0 is equivalent to its possessing a non-singular vector field. --Dan ________________________________________________________________ * If I had to guess, I might venture that this "constant volume" means that for all Riemannian metrics on the manifold that are compatible with the affine structure -- if this even makes sense -- the corresponding volume forms are just constant positive multiples of each other. Fred wrote: << See http://en.wikipedia.org/wiki/Affine_manifold for a number of other such --- "Geometry of affine manifolds is essentially a network of longstanding conjectures; most of them proven in low dimension and some other special cases." On 3/11/11, Dan Asimov <dasimov@earthlink.net> wrote: << Speaking of affine, there's a famous conjecture in differential geometry (by Shiing-Shen Chern): Define an *affine manifold* to be one having an atlas all of whose transition functions are affine. For example, R^n / x ~ 2x is an affine manifold that's topologically the product of spheres S^1 x S^(n-1). CONJECTURE: The Euler characteristic of any compact affine manifold is equal to 0. This has remained unresolved for over 50 years.
________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx
The choice of nomenclature suggests that "constant volume" has some connection with the theorem that an affine transformation of n-space changes the n-content (of every simplex) by a constant ratio. WFL On 3/11/11, Dan Asimov <dasimov@earthlink.net> wrote:
That Wikipedia article mentions two other conjectures beside the one I cited. The first one refers to affine manifolds "of constant volume" -- a notion I don't understand.* And which as far as I can tell is found nowhere in the reference cited (M. Hirsch & W. Thurston, Annals, 1975)
The third conjecture strikes me as highly technical.
But a manifold's having Euler characteristic = 0 is equivalent to its possessing a non-singular vector field.
--Dan ________________________________________________________________ * If I had to guess, I might venture that this "constant volume" means that for all Riemannian metrics on the manifold that are compatible with the affine structure -- if this even makes sense -- the corresponding volume forms are just constant positive multiples of each other.
Fred wrote: << See http://en.wikipedia.org/wiki/Affine_manifold for a number of other such --- "Geometry of affine manifolds is essentially a network of longstanding conjectures; most of them proven in low dimension and some other special cases."
On 3/11/11, Dan Asimov <dasimov@earthlink.net> wrote: << Speaking of affine, there's a famous conjecture in differential geometry (by Shiing-Shen Chern):
Define an *affine manifold* to be one having an atlas all of whose transition functions are affine.
For example, R^n / x ~ 2x is an affine manifold that's topologically the product of spheres S^1 x S^(n-1).
CONJECTURE: The Euler characteristic of any compact affine manifold is equal to 0.
This has remained unresolved for over 50 years.
________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx
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