I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ...(adsbygoogle = window.adsbygoogle || []).push({});

I am currently focussed on Chapter 3: Advanced Calculus ... and in particular I am studying Section 3.3 Geometric Sets and Subspaces of [itex]T_p ( \mathbb{R}^n )[/itex] ...

I need help with a basic aspect of Example 3.3.7 ...

Example 3.3.7 reads as follows:

Question 1

In the above text we read:

" ... ... Let [itex]p = ( x_0, y_0, z_0 ) \in S[/itex] i.e. [itex]2 x_0 - 3 y_0 - z_0 = 0[/itex]. ... ... "

... BUT ... as I read the example ... ... we have that [itex]2 x_0 - 3 y_0 - z_0 = 0[/itex] is the equation of [itex]c(t)[/itex] at [itex]( x_0, y_0, z_0 )[/itex] ... AND ... again as I see it ... this is not all of [itex]S[/itex] as [itex]c[/itex] maps [itex]I[/itex] into [itex]S[/itex] ... thus a general point [itex]p = ( x_0, y_0, z_0 ) \in S[/itex] may not satisfy the equation as it may not be in the range of [itex]c[/itex] ...

(I hope I have made my question clear ..)Can some please clarify my issue with the example... ?

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*** EDIT ***

After some reflection I now feel that all points in [itex]S = \phi (U) = ( u, v, 2u - 3v )[/itex] satisfy the equation [itex]2x - 3y - z = 0[/itex] ... so a particular point [itex]p = ( x_0, y_0, z_0)[/itex] obviously satisfies [itex]2x_0 - 3y_0 - z_0 = 0[/itex] ... is that right ... ?

Please let me know if my edit is correct ...

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Question 2

In the above text we read:

" ... ... This discussion shows that for all [itex]p \in S[/itex],

[itex]T_p (S) = \{ (a, b, c)_p \ \ | \ \ 2a - 3b - c = 0 \} \subset T_p ( \mathbb{R}^3 ) [/itex]

... ... ... ... "

Now, it seems that vectors at [itex]p = (x_0, y_0, z_0)[/itex] that have components [itex]a, b, c[/itex] respectively which obey the equation, [itex]2a - 3b - c =0[/itex] are (I think???) in [itex]S[/itex] ... ... so this would mean that [itex]T_p(S)[/itex] is a subset of [itex]S[/itex] ...Is that correct ...?

Question 3

Does anyone know of any books with a simple approach to tangent spaces replete with a number of worked/computational exercises ...?

Hope someone can help with the above questions ...

Peter

I have made a simple diagram of my understanding of the mappings involved ... as follows ... ...

Is the above diagram a correct representation of the mappings involved?

To help to give some of the context and some explanation of the theory and notation relevant to the above I am providing McInerney's introduction to Section 3.3 as follows:

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# Geometric Sets and Tangent Subspaces - McInnerney, Example 3

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