## Isometries in dimension 4?

It occurs to me that I’ve never thought about higher-dimensional generalizations of the two theorems contained in the previous post, say even in dimension 4. I’m sure it’s well-known in some form, but by thinking about it, you might even find a nice new description! To be precise, we are asking for a natural description of isometries of determinant 1 on R^4. One can think immediately of some families, like the ones that leave a line fixed and then rotate inside the three-space orthogonal to it. But then, I guess such a transformation is actually leaving a whole *plane* fixed and rotates inside the orthogonal complement plane. One could then compose two such rotations along two different planes. Hmm. I have a feeling the answer is a version of some theorem in standard Lie theory, but I can’t quite see it….Even so, you shouldn’t hesitate to think it through using your own intuition.

MK

1. Posted December 16, 2007 at 1:46 am | Permalink | Reply

I get the feeling I shouldn’t give too much away, but there’s a nice description of a general isometry of determinant 1 on R^4 as the composite of rotations in two orthogonal planes.

And, once you know this result and the similar result for R^3, it’s not hard to guess the general result for R^n. Note that the even and odd dimensions work somewhat differently.

Anyone impatient for more details can look at page 6 here… but it’s probably more fun to figure this stuff out oneself!

2. Posted December 16, 2007 at 3:21 pm | Permalink | Reply

I did want to give students a chance to think it through! But reading John’s article is certainly as instructive, so go ahead.

MK