I've asked this question on StackExchange, but until now, there is no answer or clue. I hope I can get help here.

When I read ' A course in differential geometry & # 39; & # 39; from Klingenberg, I can not be sure that the Frenet setting defined in this book is independent of the choice of the parameter of a curve.

As a result, the definition of curvature of the curve in this book confuses me.

Let $ I $ to be an interval in $ mathbb {R} $, and $ c: I rightarrow mathbb {R} ^ n $ to be a smooth card.

We consider $ c $ as a parameter of a smooth curve $ C $, and we always assume that derivatives $ c (t), c & # 39; & # 39; (t), dots c ^ {(n-1)} (t) $ are linearly independent.

According to proposition 1.2.2 of this book, there is a single Frenet framework $ e_1, dots, e_n $ along $ C $ such as (1) $ c (t), c & # 39; & # 39; (t), dots, c ^ {(k)} (t) $ and $ e_1 (t), dots, e_k (t) $ have the same orientation for $ 1 the k the n-1 $and that (2) $ e_1 (t), dots, e_n (t) $ has positive orientation.

Using this special Frenet framework, we can calculate the coefficient functions $ omega_ {ij}: I rightarrow mathbb {R} $ such as $ e_i (t) = sum omega_ {ij} (t) e_j (t) $ for all $ t in I $.

And in definition 1.3.3 of this book, the $ i $curvature $ kappa_i $ of $ C $ is defined to be the quotient

$$ kappa_i (t) = frac { omega_ {i, i + 1} (t)} {| c (t) | $$

Now, if we have another interval $ tilde {I} $ and another smooth card $ tilde {c}: tilde {I} rightarrow mathbb {R} ^ n $, as well as a diffeomorphism $ varphi: I rightarrow tilde {I} $ with $ varphi & # 39; (t)> 0 $then we can consider $ tilde {c} $ as another parameter of the same curve $ C $.

By mathematical induction, the linear independence of $ c (t), c & # 39; & # 39; (t), dots, c ^ {(n-1)} (t) $ implies the linear independence of $ tilde {c} ( tilde {t}), tilde {c} & # 39; & # 39; ( tilde {t}), dots, tilde {c} ^ {(n-1)} ( tilde {t}) $.

Proposition 1.2.2 can therefore give another frame of Frenet, which allows us to calculate the new coefficient functions $ tilde { omega} _ {ij}: tilde {I} rightarrow mathbb {R} $, and define new curvatures $ tilde { kappa} _i ( tilde {t}) $.

I do not know very well the differential geometry, but I think that the curvatures of a fixed curve should be independent of the choice of its parameter.

That's, we should have $ kappa_i = tilde { kappa} _i circ varphi $to know $ kappa_i (t) = tilde { kappa} _i ( varphi (t)) $ for all $ t in I $.

(Revisit that $ varphi: I rightarrow tilde {I} $ is defined as the change of parameters.)

But how to prove it?

The "evidence" in Klingenberg's book muddles me a lot.

Part (ii) of Proposition 1.3.2 tells us that if the change of variables $ varphi $ is given as above, and if we leave $ tilde {e} _i = e_i circ varphi $then the new $ tilde { kappa} _i $ calculated by the $ tilde {e} _i $is equal to the old $ kappa_i $ calculated by the $ e_i $& # 39; s (The proof is quite simple.)

However, these $ tilde {e} _i $& # 39; s are not calculated from the way the Frenet framework is defined in Proposition 1.2.2, that is, they are not computed by Schmidt orthogonalization of the derivatives $ tilde {c} & # 39; tilde {c} & # 39; & # 39 ;, dots, tilde {c} ^ {(n-1)} $.

They are just given by $ tilde {e} _i = e_i circ varphi $and even Klingenberg himself does not use the name & # 39; & # 39; Frenet frame & # 39; in proposal 1.3.2.

My question is whether these $ tilde {e} _i = e_i circ varphi $ are simply equal to the Frenet frame associated with the new parameter $ tilde {c}: tilde {I} rightarrow mathbb {R} $, that is, the result of the Schmidt orthogonalization of $ tilde {c} ( varphi (t)), tilde {c} & # 39; & # 39; ( varphi (t)), dots, tilde {c} ^ {(n-1)} ( varphi (t)) $ is just the $ e_1 circ varphi (t), points, e_n circ varphi (t) $?

If the answer is YES, how can we prove it?

For $ e_1 $ and $ tilde {e} _1 $this is easy by the rule of the chain.

But in general, it seems quite complex, because we have to use the formula of Faà di Bruno to calculate the higher derivatives, and the formula of Faà di Bruno is quite complex.

(I tried very very hard but I failed.)

If the answer is NO, then, to show that the definition of curvatures $ kappa_i $ are independent of the choice of parameters, we must calculate the true Frenet framework under the new parameter.

But the calculations seem rather complex for the same reason.

Thank you for your attention.