Talk:Hopf fibration

I've been looking for an explanation of why the circles of the Hopf fibration become linked. This is a request for someone more knowledgeable to fill in this missing information - Gauge 17:51, 2 Apr 2005 (UTC)

natural metric?

This is a somewhat flaky question, but ... I'm wondering if there's a "natural" metric associated with a Hopf fibration. The "natural" metric on CP^n is the Fubini-Study metric, which is identical to the ordinary metric on the two-sphere for CP^1. I can certainly pullback the metric on S^2 to define a metric on S^3, but I'm wondering how "natural" this really is, if it has any interesting non-intuitive or enligtening properties.

For example, if I envision S^3 as the EUcliden space R^3 that we live in, with an extra point at infinity, then the Hopf fibration fills this space with non-intersection circles (as illstrated by the "keyring fibration" photo). Each circle has a center ... what is the density of the distribution of the centers of these circles in R^3, (assuming a uniform density on S^2)? Are the centers of these circles always confined to a plane? What is the distribution on the plane? Uniform? Gaussian? Each circle defines a direction (the normal to the plane containing the circle). What is the distribution of these directions? linas 16:23, 26 June 2006 (UTC)[reply]

When I think of a metric on S^3 using the Hopf fibration, I think of the Berger spheres. That article needs clean-up, by the way. --Horoball 22:12, 7 October 2007 (UTC)[reply]

Dumbing down

While I sympathize with the aim of beginning articles with accessible language, the claim that "the Hopf bundle (or Hopf fibration) … is a partition of a 3-dimensional hypersphere into circles" misrepresents the essential mathematics.

Yes, a fiber bundle has fibers, but the topological relationship between the base space and the total space through the projection map is what makes it important. In particular, if we look at the inverse image of a neighborhood in the base, that portion of the bundle looks like a product of the neighborhood and the fiber space. This "local product space" structure is what allows us to do, say, path lifting.

Better pictures of the Hopf bundle suggest this topology by showing nested tori, not just circles. Some of the earliest computer graphics instances are the work of Thomas Banchoff, whose "flat torus" is the inverse image of a circle of S². And he shows circle geometry, not just topology, because the image uses stereographic projection from S³ to R³. A visualization of the entire bundle, not just one torus, can be found at the Hopf Topology Archive. Follow the link from the main page to see an image using colors on both S³ and S², and other strategems, to reveal structure. (It is also found in the SIGGRAPH 94 Art and Design Slide Set, and in Graphics Gems IV.) In this one the circles are only topological, but are confined to a finite ball.

The (pre-existing) keyrings "model" in the picture leading the article is as unhelpful as the "partition" prose, though it has other appeal. (The better images I mentioned cannot be used because of copyright.) Also, I'm afraid the "One topological model" sentence is a move in the wrong direction, especially for the lay reader, for whom it will be gibberish.

So, care to try again? --KSmrq^T 22:20, 8 October 2007 (UTC)[reply]

I largely agree, but haven't fixed this yet. Meanwhile, KSmrq, I know you are a whizz with SVGs. Do you think you could produce one? Homotopy groups of spheres also really needs a better lead image. Geometry guy 22:34, 8 October 2007 (UTC)[reply]

I'm happy to have my prose dissected and improved. I'd like to defend the "one topological model" part, though: the point was to describe the bundle in terms lay readers might hope to be able to visualize (circles in the one-point completion of 3d) rather than leaving them with the impression that as an object involving abstract 3-manifolds it is unvisualizable. Similarly, while the local product structure is essential to the mathematical content, I don't think it's essential to a lead that gives lay readers some idea of what this is about. —David Eppstein 23:34, 8 October 2007 (UTC)[reply]

Now it's your turn to critique or improve! I made a number of revisions to the intro, with mixed results. The first paragraph says more and says it better, I hope. The "one topological model" portion really didn't work for me there, so I moved it and rewrote it. I'm not thrilled with diving into notation and technicalities in the second paragraph. It happened because my imagination failed me: how do we describe the local product structure colloquially? A product space is an easy idea, but not one the lay reader knows. And a local product? Argh. I could do it in a paragraph, but not in a sentence. So I moved up some material that was already there, and improvised. The implications need expanding, but by then I was tired of losing the wrestling match. While I was at it, I switched the references to {{citation}} form so I could get the automatic links from {{harv}}, and expanded them a little.

As for an image: While tinkering with Villarceau circles several months ago I began playing with some 3D renderings, just to show the nested torus idea of stereographic projection, using transparency. If I'm doing the Hopf fibration, I want S² in the picture as well. No way would I tackle this in SVG! PostScript maybe (see Casselman); but even that would be quite the challenge. With modern graphics cards, the really cool approach would be interactive 3D graphics, but Wikipedia doesn't support that. (The chemists must really chafe.) --KSmrq^T 12:47, 10 October 2007 (UTC)[reply]

naming

Why is this article at "Hopf bundle" instead of "Hopf fibration" anyway? --Horoball 17:39, 9 October 2007 (UTC)[reply]

I have been asking myself the same question since I saw it, so I've now moved the page. Geometry guy 17:45, 9 October 2007 (UTC)[reply]