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what ain't a manifold at this point
by termiefoo 8y ago
what ain't a manifold at this point
- username90 8y agoThings with edges, like disks and finite lines.
- srean 8y agoNothing wrong with them, they can be manifolds.
- deleted 8y ago[deleted]
- math_and_stuff 8y agoBoth of those are typically referred to as manifolds with boundary. The setting of Stokes Theorem is often manifolds with boundary.
- hansen 8y agoMathematicians messed up here… A manifold with boundary is not a manifold. But a manifold is a manifold with boundary (the empty set).
- deleted 8y ago[deleted]
- Topolomancer 8y agoI thought a manifold with a boundary would still be a manifold, but its boundary has to satisfy a dimensionality condition. For example, the 2D disk is a 2-manifold with a 1-dimensional boundary. Strictly speaking, this is a _topological manifold with a boundary_, though.
- JadeNB 8y ago> Mathematicians messed up here… A manifold with boundary is not a manifold. But a manifold is a manifold with boundary (the empty set). It depends on which mathematicians! Plenty of differential geometers allow manifolds to have boundary, and say "closed manifold" (https://en.wikipedia.org/wiki/Closed_manifold https://en.wikipedia.org/wiki/Closed_manifold) to emphasise when they are dealing with a (compact) manifold without boundary (or, as you point out, really a manifold whose boundary is empty).
- deleted 8y ago[deleted]
- srean 8y agoA fractal isnt a manifold. Discrete sets are an edge case. They are considered to be 0 dimensional manifolds to make them fit, but there is a not much that manifold theory has to say about them.
- man-and-laptop 8y ago> A fractal isnt a manifold Some are. A Koch curve for instance.
- srean 8y agoI believe you are correct. Koch curve is a continuous curve so its topological dimension would be 1 (not its fractal dimension).
- JulianWasTaken 8y agoBesides the other answers, things also start to get interesting when you add some qualifiers. E.g., asking for a "differentiable manifold" now asks for manifolds with enough additional structure to do calculus on top of.
- Topolomancer 8y agoGood point; I originally wanted to talk about the Poincaré conjecture as well, but then I realized that this would make the post even longer. Do you have some ideas about other interesting topics?
- JulianWasTaken 8y agoTo be honest it's not even my field so I know a lot less than I wish I did but yeah agreed it has no shortage of interesting things to talk about :) Poincaré Conjecture is yeah interesting to explain though to be honest I vaguely recall an article in similar style to your post that explained it. Possibly something like describing gradient descent on a manifold is interesting to this audience? Or maybe a post on Flatland? Many possibilities really on good follow ups.
- amelius 8y ago> Good point Can a point be a manifold?
- Topolomancer 8y agoYes, but it's a simple one of dimension zero. There's not much to do here---every 'neighbourhood' of the point is exactly that: just the point. (not that there are different 'classes' of manifolds out there; I am not sure if the point would qualify as a Riemannian manifold, for example)
- ColinWright 8y agoA Y-shaped "line".
- badosu 8y agof(x) = 0 (x < 1, x > 1) f(0) = 1
- Sinbe 8y agoSimplest example of what not a manifold is the shape of the letter “Y”. Technically for a topological space to be a manifold, every small neighborhood around each point should look the same. But “Y” has a special point at the center.