I've recently been thinking about various problems involving two points on the surface of a unit sphere. Let's specify them with a pair of unit 3-vectors ${\bf \hat a}$ an ${\bf \hat b}$. Is there some aid to thinking about this 4-dimensional space? In particular:

  1. What is its topology? (In terms suitable for dumb engineers like me.)
  2. How do I integrate over parts of this space? The measure would derive from surface area on the original sphere but the integrands of interest are probably all simple functions the scalar ${\bf \hat a\cdot \hat b}$
  3. Is there a coordinate space or other model that makes it easy to deal with questions like (1) and (2).
  • $\begingroup$ I don't see how you define a 4-D space with two vectors in the 3-sphere. Would you explain? $\endgroup$
    – user99680
    Jun 7, 2014 at 5:51
  • 1
    $\begingroup$ The 3-sphere is a two dimensional space (just a surface). Two independent points on a two dimensional space gives four dimensions in total. $\endgroup$ Jun 7, 2014 at 6:31
  • $\begingroup$ That's not how dimensions work. $\endgroup$ Jun 7, 2014 at 7:21
  • $\begingroup$ @user99680: He is talking about two points in the 2-sphere, not in the 3-sphere. $\endgroup$
    – Lee Mosher
    Jun 7, 2014 at 12:48

1 Answer 1


This is $S^2 \times S^2$; you can give it the product topology, so that a basis for the topology is given by $\{U \times V \mid U, V \text{ open } \subset S^2\}$. In order to think about 1 and 2, it might help to think of it as a manifold; you can use polar coordinates to homeomorphically biject open sets to open sets of $\mathbb{R}^4$, and you can integrate by using the change of coordinates formula:


  • $\begingroup$ That's a very nice first answer. Keep it up! $\endgroup$
    – M. Vinay
    Jun 7, 2014 at 8:45

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.