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Mathematics
Differential Geometry
Elementary Differential Geometry: Cone Not a Surface - Exercise Problem
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[QUOTE="paluskar, post: 3614766, member: 360566"] [b]cone as surface...[/b] In Barrett O'Neill's Elementary Differential Geometry book...he says that the cone M:[itex]x^{2}[/itex] + [itex]y^{2}[/itex]=[itex]z^{2}[/itex] is not a surface in that there exists a point [B]p[/B] in M such that there exists no proper patch in M which can cover a neighbourhood of [B]p[/B] in M Intuitively I realize that this is point is the apex...here [itex]\left(0,0,0\right)[/itex] but how would the patches for the rest of points be??...and what goes wrong with the patch at the apex of the cone?? Note: A PATCH IS A 1-1 REGULAR FUNCTION X:D→[itex]ℝ^{3}[/itex]..D open in [itex]ℝ^{2}[/itex] This is an exercise problem. [/QUOTE]
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Differential Geometry
Elementary Differential Geometry: Cone Not a Surface - Exercise Problem
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