[SOLVED] show SO(3) a manifold inside R^6 and prove SO(3) embeds in R^5
have done some research, and here is what I know;
to show SO(3) a manifold "inside" R^6(am guessing inside means a submanifold), the hint was that only two vectors are needed to describe the tangent space of the 2-sphere, ie if given a radial vector, and a tangent vector, the other orthonormal vector is then completely specified. more specifically;
the Jacobian takes tangent spaces to matrices smoothly,
so am trying to write the diffeo from the tangent space to the sphere S^2(T_xS^2)
into SO(3). This was from the hint. It seems the jacobian would fill this bill.
Also found that RP^3 can be constructed from the unit Ball, and that can then construct a diffeo from RP^3 to SO(3)(take unit Ball in above to have radius Pi), but don't see how this would help to show a submanifold,
when I look at the definition below;
Starting from the definition of a submanifold;
A C^infty (infinitely differentiable) manifold is said to be a submanifold of a C^infty manifold M^' if M is a subset of M^' and the identity map of M into M^' is an embedding. Well clearly SO(3) is a subset of R^6, so need show the identity map from SO(3) into R^6 is an embedding. Here it seems that one could just add zeros to the vectors in SO(3), but this must be too simple.
An embedding is a proper immersion, and proper means only that the inclusion
map is closed(I think). Well, not too sure how to tie all this together.
To show SO(3) embeds in R^5, must exhibit an embedding, and this wouold be harder than above, not sure how to proceed.
thanks in advance for any help, suggestions you may be able to render.