Re: Proof: SO(3) diffeomorphic to RP^3


"Lee Rudolph" <lrudolph@xxxxxxxxx> schrieb im Newsbeitrag
news:eqnf97$bkp$1@xxxxxxxxxxxxxxxxxxx
"Pat Eitsch" <pfilz0@xxxxxxxxxxx> writes:


"Lee Rudolph" <lrudolph@xxxxxxxxx> schrieb im Newsbeitrag
news:eqn3sr$pmc$1@xxxxxxxxxxxxxxxxxxx
"Pat Eitsch" <pfilz0@xxxxxxxxxxx> writes:

Hi y'all,

I have not posted anything in this newsgroup before and I have this
proof
of
the above statement which I constructed myself but am not sure whether
it's
correct. Now my first question is, can I post something like that in
this
forum or would nobody help me?

Please do post it.

Ok, here it is:

[proof, omitted]

That appears to me to be perfectly correct. To my taste, it's
overly complicated *as a proof of the stated result*, but of course
it proves more than that and is full of interesting things in its
own right. An alternative proof (of just the stated result) goes
like this. We can map the closed unit ball in (oriented) R^3 onto
SO(3) by mapping the vector v of length |v|, 0 < |v| <= 1, to the
rotation around the axis Rv through the counterclockwise angle
|v|pi, and mapping 0 to the identity. It's trivial to check that
this map is continuous (also at 0) and onto, and that it fails to
be injective only insofar as it maps v and -v to the same rotation.
Thus SO(3) is homeomorphic to the quotient space of the unit ball
under identification of antipodal boundary points, which is RP^3.
Since you were willing to invoke "homeomorphism implies diffeomorphism
for 3-manifolds" in your proof, I'll invoke it here too. (But in both
your proof and mine, with more care the homeomorphism and its inverse
could be shown to be differentiable.)

Lee Rudolph

Thanks for taking the time to look through it. Escpecially since - in my
opinion - matrices and such are hardly legible in plain text. Maybe next
time I should use latex and attach the pdf file. I have seen a few proofs
similar to the one you stated and I like its brevity, but I wanted to carry
out the calculations to see how you can use quaternions to represent
rotations in R^3, because I used to think you can't use them for anything at
all.

Regards,
Pat


.

Relevant Pages

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