Projection or Inclusion?

1. Aug 29, 2010

FreHam

Hi,

Suppose I have a space X with coordinates (x,y,z) and a space Y with coordinates (x,y,z,t), so that dim(Y)=dim(X)+1.

What is the difference between the projection (x,y,z,t)->(x,y,z) and the inclusion (x,y,z)->(x,y,z,t)? Are they each others inverses? Especially if x=x(t), y=y(t) and z=z(t)?

I'm really stuck somehow.

Cheers,

Fred.

2. Aug 29, 2010

HallsofIvy

I'm not sure what you mean by "the difference between" (x, y, z, t)-> (x, y, z) and (x, y, z)->(x, y, z, t). Obviously one is from R4 to R3 and the other is from R3 to R4.

However, the second one is not well defined since there is no way of knowing what t to append to (x, y, z). No, they are not inverse to one another. There exist an infinite number of (x, y, z, t) that map to the same (x, y, z) so the function is not "one to one" and does not have an inverse.

If you specify that x, y, and z are functions of t, you still have a problem- (x(t), y(t), z(t), t)- > (x, y, z) does NOT map R4 to R3, it maps a one-dimensional subset of R4 onto a one dimensional subset of R3. Also, trying to go from (x, y, z) to (x(t), y(t), z(t), t), a given triple, (x, y, z) may contain x, y, z, values that correspond to different values of t.

3. Aug 29, 2010

FreHam

Is there then a possibility to define a map from the lower-dimensional space to the higher one? I'm basically considering time-dependent systems on a (2n+1)-dimensional contact manifold T*Q x R, and I want to include/embed (don't know what term to use) in a (2n+2)-dimensional symplectic manifold. The coordinates on T*Q x R are (q,p,t) and on the symplectic one (q,p,q',p'), where the primes denote some additional coordinates. Basically, q'=t but of some new time parameter, say s.

4. Aug 30, 2010

HallsofIvy

Yes, but not "onto". You could have a linear map that maps a space of dimension n to an n-dimensional subspace of a space of dimension m (n< m). That maps the n dimensional space into the m dimensional space.

5. Aug 31, 2010

FreHam

So, do I get this right: you can only call a map from X to Y an inclusion if dim(X)=dim(Y)? How would you call a map from a lower-dimensional space to a higher-dimensional space? Embedding? Immersion? Inclusion? ...?

I'm not sure whether my case would actually be a symplectization of a contact manifold... hmm.