Is Surface Smoothness Merely a Product of Its Parametrization?

[quasar987]

According to Presley (Elementary differential geometry),

"A smooth surface is a surface $\mathbf{\sigma}$ whose atlas consists of regular surface patches."

(The atlas of a surface is a collection of homeomorphisms that "cover" it. A surface patch is just another word for an homeomorphism in the atlas. Finally, a surface patch is regular if it is smooth and its first partial derivatives are linearly independent at all points (u,v) of its domain.)

Generally, there are several possible distinct atlases for a given surface. A priori, I see no reason to say that if a surface is smooth under some atlas, it is under every atlas.

So, is it really so that smoothness is not an intrinsic property of surfaces, but rather a "bonus" that comes with a proper choice of parametrization? (much like regularness of a curve is a property of the parametrization, not of the trace itself)

 

[AKG]

Yes, if I recall correctly, it depends on the atlas.

 

[quasar987]

But then there is a thm latter on that goes

Let S be a subset of R^3 with the following property: for each point P in S, there is an open subset W of R^3 containging P and a smooth function f:W-->R such that
(i) S $\cap$ W={(x,y,z) in W: f(x,y,z)=c};
(ii) the partial derivatives f_x, f_y and f_z do not all vanish at P.
Then S is a smooth surface.

The statement "S is a smooth surface" doesn't make sense on its own! One that does however is "Then there is an atlas for which S is smooth". The way the author proves this thm is by finding an atlas of regular patches, i.e. one for which S is smooth. So it would appear that this is all the author meant by "S is smooth". Unless, we've overlooked something and smoothness really is an intrinsic property!

 

[mathwonk]

first you have to define smoothness. when you do, you will see that it depends eitheron the parametrization by surfqce patches, or on the embedding, which also tacitly ssumes that projection on some axes is an allowable family of paTCHES.

 

[mathwonk]

BY THE WAY, the imprecision of those descriptions made me assume you were reading a physics book and not a math book. gosh. i recommend geting a better book. like do carmo.or spivak, or shifrins web notes, or if those are too advanced, maybe barett o'neill for a baby book.

 

[quasar987]

It's a maths book. Elementary Geometry by Andrew Pressley.

But thanks for the books recommendation!

Which spivak book are you referring to?