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Talk:Hopf fibration

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This is an old revision of this page, as edited by 192.35.232.241 (talk) at 16:22, 26 June 2006 (natural metric?). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

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? 192.35.232.241 16:22, 26 June 2006 (UTC)[reply]

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