How to reason about two points on the unit sphere.

Question

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).

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:

https://en.wikipedia.org/wiki/Integration_by_substitution#Substitution_for_multiple_variables