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 Definition 3.3.5 ...

Definition 3.3.5 reads as follows:

In the above definition we find the following:

" ... ... Here, when we say that a parametrized curve [itex]c \ : \ I \longrightarrow S[/itex] is smooth, we mean that there is a smooth function [itex]\tilde{c} \ : \ I \longrightarrow U[/itex] such that [itex]c = \phi \circ \tilde{c} [/itex] ... ... "

My question is as follows:

Why do we need to bother defining [itex]\tilde{c}[/itex] ... the codomain of [itex]c[/itex] is defined as [itex]S[/itex] ... so we surely only need to stipulate that [itex]c[/itex] is continuously differentiable or [itex]C^1[/itex] ... that is the usual definition of 'smooth' so why isn't this enough ...

... ... so, my question is then, why do we bother defining [itex]\tilde{c}[/itex] and then go on to consider the composite function [itex]c = \phi \circ \tilde{c}[/itex] ... ?

Hope someone can help ...

Peter

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To give the context for McInerney's approach to this definition I am providing the introduction to Section 3.3 as follows:

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# Tangent Spaces of Parametrized Sets - McInerney, Defn 3.3.5

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