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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeJan 22nd 2010

    Urs created Frechet manifold, so I created Frechet space. (We violated the naming conventions too, but I guess it's OK since we have the redirects in.)

    • CommentRowNumber2.
    • CommentAuthorTim van Beek
    • CommentTimeApr 28th 2011
    Over at Azimuth we've been discussing partial differential equations for hydrodynamics, and John pointed out the work done by Arno'ld et. alt. about characterizing the flow of ideal fluids (incompressible, inviscuous, homogenous) as flowing along the geodesics on the Fréchet manifold of volume preserving diffeomorphisms of the fluid domain manifold.

    I don't know much about Riemannian geometry on Fréchet manifolds besides that there isn't a full general theory, because of the problem of duality (duals of Fréchet TVS need not be Fréchet TVS) and much more because of the problem that finite dimensional existence and uniqueness theorems of solutions of ODE fail in infinte dimensions.

    Although my main objective at this point is an implementation of a numerical algorithm for a simple toy problem involving the Burgers equation, I'm interested in understanding the general theory, too, so maybe I'll expand the Fréchet manifold entry along the way.
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2011

    Hi Tim, good to see you again! It was a pity when you left us, you did some great work here on the Lab.

    • CommentRowNumber4.
    • CommentAuthorAndrew Stacey
    • CommentTimeApr 28th 2011

    One can sometimes consider co Riemannian geometry on Frechet manifolds. It can solve a few problems that Riemannian structures can’t.

    • CommentRowNumber5.
    • CommentAuthorTim van Beek
    • CommentTimeApr 28th 2011
    Hi Urs, glad to hear that :-)

    Hi Andrew, I had hoped that you'd have a comment or two on this subject :-)
    • CommentRowNumber6.
    • CommentAuthorAndrew Stacey
    • CommentTimeApr 28th 2011

    Tim, I probably have many! Indeed, this co-Riemannian stuff is something that I’ve had going round for a while and although I have plenty of ideas as to how it might be used, I’ve not done as much on it as I ought partly because I’m not sure which to concentrate on. If you can describe a situation where it might be useful, I’d be happy to get my teeth in to it and rip it to shreds.