# Cone as submanifold of R^3

#### Pietjuh

For a homework assignment i was asked to proof that the positive cone {x^2 + y^2 = z^2, z>= 0} cannot be a submanifold of any dimension of R^3.

It apparently goes wrong at the origin. I guess it's because you can't really speak of a tangent space at that point. So I tried to prove by contradiction you can't have a tangent space at that point. But I couldn't really arrive at a contradiction

Could someone give me a hint?

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#### amcavoy

Pietjuh said:
For a homework assignment i was asked to proof that the positive cone {x^2 + y^2 = z^2, z>= 0} cannot be a submanifold of any dimension of R^3.
It apparently goes wrong at the origin. I guess it's because you can't really speak of a tangent space at that point. So I tried to prove by contradiction you can't have a tangent space at that point. But I couldn't really arrive at a contradiction
Could someone give me a hint?
Try finding an equation for a tangent plane at the origin. That's the only thing I can think of (which is basically what you already had in mind!)

Alex

"Cone as submanifold of R^3"

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