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# Geometry problem (tangent plane) watch

1. I can't get my head around many of the things surrounding tangent planes: coordinate charts, atlases, Riemannian metrics... which probably isn't making my life very easy here. Anyway, I'm having a bit of trouble with showing that the tangent plane to at a point is precisely the set of vectors in the plane normal to the vector .

I've started by supposing that and considering stereographic projection . Then the inverse map is given by . So suppose . If we write then the tangent plane is given by .

So I worked out the partial derivatives to find that

So suppose and denote by . Then:

And hence, taking the inner product with respect to the standard metric on :

...which is not equal to zero for all , as it should be.

Now if you've read this far, I congratulate and thank you, and ask: can you see where I've gone wrong?

Thanks all
2. It's easier to work with the fact that the tangent plane at the image of a point (a,b) is given by the span of dsigma/du (a,b) and dsigma/dv (a,b), where sigma is the map from your open subset of R^2 to the open subset of your surface.

I kind of agree - the definition of a co-ordinate chart is 'the wrong way round' to me - it should really be a function from R^2 to R^3, not the way round we've defined it.
3. (Original post by nuodai)
...
Can't comment on most of it, as it's a bit beyond me; fortunately however you've worked out the inner product incorrectly (there are a few errors, at least before you edited it, i've not checked again).

I get the numerator to come to zero, bear with me as there are 14 terms to type out.

I may have re-ordered slightly.

Edit: You do seem to have too many terms in your expression.
4. (Original post by around)
It's easier to work with the fact that the tangent plane at the image of a point (a,b) is given by the span of dsigma/du (a,b) and dsigma/dv (a,b), where sigma is the map from your open subset of R^2 to the open subset of your surface.
Ah yes, I forgot that definition. I might re-do the question that way if I get the chance.

(Original post by ghostwalker)
Can't comment on most of it, as it's a bit beyond me; fortunately however you've worked out the dot product incorrectly (there are a few errors, at least before you edited it, i've not checked again).

I get the numerator to come to zero, bear with me as there are 14 terms to type out.

I may have re-ordered slightly.

Edit: You do seem to have too many terms in your expression.
Funnily enough, on paper I only have 14 terms; the error arose when I copied up the question. However, it seems my error here is my inability to realise that

Thanks both for your help.
5. (Original post by around)
I kind of agree - the definition of a co-ordinate chart is 'the wrong way round' to me - it should really be a function from R^2 to R^3, not the way round we've defined it.
That's called a parametrisation. A chart on a k-manifold M is a diffeomorphism where and are open. Note it doesn't have to be defined on an open subset of ambient space!

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