# I Differential structure on a half-cone

#### cianfa72

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 $\mathbb R^3$.

#### cianfa72

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?

#### cianfa72

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.

#### cianfa72

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.

#### cianfa72

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.

#### cianfa72

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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#### cianfa72

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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