Differential structure on a half-cone

In summary: 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 ?Yes, that is correct.Yes, that is correct.
  • #1
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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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  • #2
Why is that a problem? The structure is not induced from the cone as a subset of ##\mathbb R^3##.
 
  • #3
martinbn said:
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
cianfa72 said:
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
martinbn said:
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
But what is that one chart? The embedding in ##\mathbb R^3## is not a chart.
 
  • #7
martinbn said:
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
No, a chart has to map it to ##\mathbb R^2##. Besides the image in ##\mathbb R^3## is not an open set.
 
  • #9
martinbn said:
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
By the definition of a chart the image has to be an open subset.
 

1. What is a differential structure?

A differential structure is a mathematical concept used in differential geometry and topology to describe smoothness and continuity of functions on a space. It is a collection of charts (coordinate systems) that cover the entire space and allow for the definition of smooth functions.

2. What is a half-cone?

A half-cone is a geometric shape that is formed by rotating a straight line segment around one of its endpoints. It is half of a cone, which is a three-dimensional shape with a circular base and a curved surface that tapers up to a point.

3. How is a differential structure defined on a half-cone?

A differential structure on a half-cone is defined by a set of charts that cover the surface of the half-cone. These charts map points on the surface to points in a coordinate system, allowing for the definition of smooth functions on the half-cone.

4. What are the properties of a differential structure on a half-cone?

A differential structure on a half-cone has the properties of being smooth and continuous, meaning that functions defined on the surface of the half-cone can be differentiated and integrated. It also allows for the calculation of tangent vectors and curvature on the surface.

5. How is a differential structure on a half-cone used in science?

In science, a differential structure on a half-cone can be used to model and analyze physical phenomena that have a curved, tapered shape. It can also be used in the study of differential equations and the geometry of surfaces. Additionally, it has applications in fields such as physics, engineering, and computer graphics.

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