Projection and Inclusion in Higher-Dimensional Spaces: What's the Difference?

[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.

 

[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.

 

[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.

 

[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.

 

[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.