Interpreting General Principle: What Does It Mean?

[quasar987]

From the book:

General principle: We can define a property of any smooth surface provided we can define it for any surface patch in such a way that it is unchanged when the patch is reparametrized.

That so doesn't sound right. Say I have a smooth surface S and an atlas of regular surface patches. I then define a property X of a surface patch and verify that this property is independant of a change of parametrization. According to the principle, I have unambiguously defined property X for the smooth surface S itself.

But say for exemple that the atlas of S is made of two surface patches #1 and #2 that map to distinct areas of S. Now suppose that according to our definition of X, surface patch #1 is X while surface patch #2 is not X. What do we say about S, is it X or not?


I must be misinterpreting the "general statement". What does it mean to you?

 

[matt grime]

If you have such an atlas, then your surface is the (disjoint) union of two components. Usually, one works with irreducible things.

 

[quasar987]

If I interpret you correctly, you're saying that it is implicit in the general statement that S can be parametrized by a unique regular surface patch?

 

[quasar987]

I don't understand what you mean by "usually one works with irreducible things". A sphere for instance, has an atlas of minimum 2 surface patches. And if one wants, say, to compute the length of a curve diping in both these surface patches, one will have no choice but to work with both patches.

A natural way to define the property of a surface IMO is to define it for a surface patch and if each surface patch of a given atlas of the surface has the property, and additionally, if the property is such that it is unchanged under reparametrization, then the surface is said to have said property.

 

[matt grime]

You specified an atlas with two disjoint surface patches, didn't you? That is what I took 'distinct' to mean. Your example of the sphere has patches that overlap.

The general principal above is the local-global principle: you get global information by patching together local information. It is a rule of thumb, and not a theorem.

You also appear to have changed your mind about it being (in)correct.

 

[quasar987]

You specified an atlas with two disjoint surface patches, didn't you? That is what I took 'distinct' to mean. Your example of the sphere has patches that overlap.

The general principal above is the local-global principle: you get global information by patching together local information. It is a rule of thumb, and not a theorem.

But what if you get two local informations that are contradictory? To take the sphere again, say you have it covered by two patches, one that exbihit property X, and the other that does not. You can't say anything about the sphere itself in this case.

You also appear to have changed your mind about it being (in)correct.

Why? I only reformulated the principle by adding the important condition that all the patches are carrier of the property. (unlike in the hypothetical exemple above)

 

[matt grime]

You appear to have used the (common) misapprehension that 'any' and 'for all' are different. They are normally the same. So when we say 'can be defined for any patch' we *do* mean 'for all patches'.

 

[quasar987]

This is what I began to suspect earlier today. But thanks for confirming that!

It is not the first time that I am convinced that 'any' and 'for all' mean different things. I hope it will be the last time. >:|

 

[mathwonk]

that principle was not stated very clearly or precisely there, but

It seems to be a clumsy way of saying that any LOCAL statement may be meaningfully stated for a manifold, provided
1) it is stated for R^n,
2) its truth or falsity is diffeomorphism invariant.e.g. if we state that "the derivative of a smooth map is non zero at a point p of a manifold", this qualifies, since change of coordinates multiplies the derivative by a non zero quantity.

to say "the derivative at p is the identity", does not qualify, since a change of coordinates can transform the derivative by any invertible linear map.

The phenomenon Matt mentions, of a local global principle, is more subtle, involving connectivity. This is alsoa useful general principle, but i think is not the one meant by the author.

this subtler principle say roughly, any smooth quantity which both exists locally and is unique locally, also exists and is unique globally, provided the manifold is connected.

for instance a continuous lift of a path to a (pointed) covering space. Locally a covering space is a homeomorphism so a lift exists. Since our spaces have chosen base points, also the lift is locally unique. thus since an interval is connected, a lift exists and is globally unique.clumsily stated or not, physicists usually know what they are talking about, once you understand them, so it is useful to try.

 

[mathwonk]

does that help?

 

[quasar987]

I'd be lying if I said I fully understood a single* sentence of your post.

I understand and agree with thr principle if it is saying that...

"We say that a smooth surface has property X if property X is defined for a surface patch and each surface patches of a given atlas of the surface has the property, and additionally, if the property is such that it is unchanged under reparametrization."


*I was going to write "any" at first but I decided to steer clear.