# Differential structure on a half-cone

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

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 ?

martinbn
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?

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 ?

martinbn
But what is that one chart? The embedding in ##\mathbb R^3## is not a chart.

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 ?

martinbn
No, a chart has to map it to ##\mathbb R^2##. Besides the image in ##\mathbb R^3## is not an open set.

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 ?

martinbn
By the definition of a chart the image has to be an open subset.

ok, take now an elliptic paraboloid ##C=\left\{x,y,x^2+y^2\right\}##. The same argument about half-cone applies for it (its inclusion in ##\mathbb R^3## is not open thus we cannot endow it with a differential structure via a map to ##\mathbb R^3##) nevertheless from, let me say, a "visually" point of view it seems a smooth surface as opposed as the half-cone.

Can you help me in understanding the difference between them ?

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Thinking again about it, I believe the difference between them is that the elliptic paraboloid as subset of ##\mathbb R^3## is an embedded submainfold while in the half-cone case a smooth embedding in ##\mathbb R^3## actually does not exist.

Just to elaborate a bit...Consider ##\mathbb R^2## and ##\mathbb R^3## with the standard differentiable structures. With the given smooth structures the map ##f:(x,y)\to(x,y,z=\sqrt {x^2+y^2})## that try to "immerse" the ##\mathbb R^2## plane in ##\mathbb R^3## is injective but actually is not an immersion https://en.wikipedia.org/wiki/Submanifold#Immersed_submanifolds because the lack of differentiability at ##(0,0)##. Because of that ##f## is precluded to be an embedding therefore the half-cone as subset of ##\mathbb R^3## is not a submainfold.

Does it make sense ? Thanks

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