quotient manifold

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Kimmy Kong

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Sep 12, 2023, 3:35:50 PMSep 12
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Dear all,
 I have problem understanding of the quotient concept. 
(I could understand a quotient manifold very basis, I guess,
for example, a square space, and any point map to straight line, then those line divide the square space. )

I want to ask a silly question, 
 if I consider that a sphere (2D) is a manifold M', 
then a Euclidean space (3D) could be the quotient manifold M, corresponding to this sphere?

or is this the opposite, 
Euclidean space (3D) a manifold M', then  sphere (2D) could be the corresponding quotient manifold

thank you very much
have a nice day


   

Nicolas Boumal

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Sep 13, 2023, 7:33:40 AMSep 13
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Hello,

If you think about a two-dimensional sphere "floating" in R³, then R³ is the "embedding" space of the sphere, and the sphere is "embedded" in R³. There are no quotients here.

To have a quotient manifold, you first need to have an "equivalence relation". For example, you could say that two points x and y in R³ \ {0} (that is, two non-zero points in R³) are "equivalent" if x = a*y  where a is a positive real number (a > 0). If you do that, then the quotient of R³ \ {0} by that equivalence relation is indeed "sort of the same as" the 2-D sphere (we can make this precise). So, in that case, we might say that the 2-D sphere is a quotient manifold of R³\{0}. But there is no meaning to that last sentence unless you specify the equivalence relation.

Best,
Nicolas

Kimmy Kong

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Sep 13, 2023, 2:56:25 PMSep 13
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 I have one more question, sorry.
Thank you very much for your reply.

1) Can  I  consider any interested manifold (e.g., Oblique manifold, Stiefel manifold, Symmetric, positive definite matrices, ..) are the quotient manifold of a Euclidean manifold ----(just I do not know what the equivalence relation is for each of them).?
I understand the embedded space, 
because
2) In your book, you introduce vertical space, and horizontal space for the quotient manifold,   It is not introduced in embedding space 
I want to ask, if I have to specify a quotient manifold, then I can use the horizontal space concept.

thank you very much.
have a nice day

Nicolas Boumal

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Sep 14, 2023, 2:52:02 AMSep 14
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1) In full generality, probably not in a useful way. 
2) Horizontal and vertical spaces make sense for quotient manifolds. For embedded submanifolds, you may be interested in tangent and normal spaces. The normal space is the orthogonal complement of the tangent space.

Best, 
Nicolas 

Kimmy Kong

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Sep 14, 2023, 4:40:59 AMSep 14
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Thank you very much, 
I understand it now. Yes, normal space is the one I am looking for. 
have a nice day

lingping
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