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Bases for Tangent Soaces and Subsaces ... McInerney Theorem 3.3.14 ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,887
Hobart, Tasmania
I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.3: Geometric Sets and Subspaces of [FONT=MathJax_Math]T[/FONT][FONT=MathJax_Math]p[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_AMS]R[/FONT][FONT=MathJax_Math]n[/FONT][FONT=MathJax_Main])[/FONT] ... ...

I need help with an aspect of the proof of Theorem 3.3.14 ... ...

Theorem 3.3.14 reads as follows:


McInerney - 1 - Theorem 3.3.14 ... ... Page 1 ... .png
McInerney - 2 - Theorem 3.3.14 ... ... Page 2 ... .png


Can someone please explain/demonstrate explicitly why \(\displaystyle (e_1)_p, (e_2)_p, \ ... \ ... \ , (e_{n-1} )_p\) are the standard basis vectors for \(\displaystyle T_p( S_f )\) ...


Help will be much appreciated ... ...

Peter


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The above post mentions Theorem 3.3.13 so I am providing text of the theorem together with a relevant definition ... as follows:


McInerney - Defn 3.3.12 & Theorem 3.3.13 ... ... Page 1 ... .png
McInerney - 2 - Defn 3.3.12 & Theorem 3.3.13 ... ... Page 2 ... .png


Hope that helps readers follow the post ...

Peter