Differential structure on a half-cone

  • #1
cianfa72
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Hi,

consider an "half-cone" represented in Euclidean space ##R^3## in cartesian coordinates ##(x,y,z)## by: $$(x,y,\sqrt {x^2+y^2})$$
It does exist an homeomorphism with ##R^2## through, for instance, the projection ##p## of the half-cone on the ##R^2## plane. You can use ##p^{-1}## to get a differential manifold structure on it.

It seems good, but I've some problem with it because of lack of tangent plane on the apex with coordinates ##(0,0,0)## when we think it living in ##R^3##.

Can you help me ? thanks
 
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Answers and Replies

  • #2
martinbn
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Why is that a problem? The structure is not induced from the cone as a subset of ##\mathbb R^3##.
 
  • #3
cianfa72
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Why is that a problem? The structure is not induced from the cone as a subset of ##\mathbb R^3##.
I try to elaborate a bit.

Consider the inclusion map ##\iota: S \mapsto \mathbb R^3## between the cone as subset ##S = \{x,y,\sqrt {x^2+y^2}\}## of ##\mathbb R^3## and ##\mathbb R^3## itself. Using ##\iota^{-1}## we can provide ##S## with the subspace topology induced by standard ##\mathbb R^3## topology.

Now I believe ##\iota: S \mapsto \mathbb R^3## is homeomorphism with its image in ##\mathbb R^3## thus this way we can provide a differential structure on the half-cone (not sure if it is actually the same as that defined by the half-cone projection on the ##\mathbb R^2## plane)

If what said is correct, I suppose the so defined differential structure on the half-cone ##S## is actually compatible with the ##\mathbb R^3## standard differential structure. Is that right ?
 
  • #4
martinbn
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Now I believe ##\iota: S \mapsto \mathbb R^3## is homeomorphism with its image in ##\mathbb R^3## thus this way we can provide a differential structure on the half-cone
How does it provide a differential structure?
 
  • #5
cianfa72
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How does it provide a differential structure?
Sorry I'm a newbie on this topic, not sure to fully understand your question.
However following the definition of differential manifold, it should be suffice to define a "compatible" atlas for the half-cone ##S## and if the atlas contains just one chart (as in this case - see before) it alone basically defines the manifold as a differentiable manifold, do you ?
 
  • #6
martinbn
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But what is that one chart? The embedding in ##\mathbb R^3## is not a chart.
 
  • #7
cianfa72
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But what is that one chart? The embedding in ##\mathbb R^3## is not a chart.
Maybe I'm wrong...but the inclusion map ##\iota: S \mapsto \mathbb R^3## does not allow to define the chart ##(S,\iota)## on ##S## itself ?
 
  • #8
martinbn
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No, a chart has to map it to ##\mathbb R^2##. Besides the image in ##\mathbb R^3## is not an open set.
 
  • #9
cianfa72
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No, a chart has to map it to ##\mathbb R^2##. Besides the image in ##\mathbb R^3## is not an open set.
Just to check if I got it correctly: the inclusion map ##\iota## is good to endow the half-cone with ##\mathbb R^3## subspace topology, however it does not define a chart also because ##S## is not an open set in ##\mathbb R^3##.
Thus for the very fact that ##S## is not open in ##\mathbb R^3## it does not exist a chart to map it on ##\mathbb R^3##, right ?
 
  • #10
martinbn
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By the definition of a chart the image has to be an open subset.
 

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