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A curve $\sigma$ is defined to be an equivalence class of parametric curve under the equivalence relation of change of parameter. For regular curves there is a canonical element of the equivalence class, i.e the parametric curve which is called the natural parametrisation, using the arc parameter.

Remark: Note that two elements of the same equivalence class don't have necessarly the same derivative.

Given a curve $\sigma$ on a surface $S$, then it is a geodesic curve for the surface if the covariant derivative of $\sigma'$, $D\sigma'\equiv0$

Now it seems to me that this definition involves the elements of an equivalence class, i.e. the parametric curves , and it is not independent from the parametrisation, this is also due to the important geodesic equation:

$\\ \sigma_j'' + \sum_{h,k=1}^2 \Gamma_{hk}^{j} \sigma_h'\sigma_k' = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j=1,2 $

Which involves the derivatives.

But also there is a property of geodesic curves, which is that they have $||\sigma'||\equiv const$, and so they are parametrised with respect to a multiple of the natural parameter.

So, my question is: To me, to be a geodesic on (say) a plane is to be a straight line, so something with a geometric description in terms of what a curve is, not its parameter, similarly to be a geodesic of a sphere is to be a maximum circle, or for a cylinder to be a geodesic to me is to be a circle,a straight line or an helix, and so on... So why here we must care about the parameter? I mean, ok if we have a geodesic by definition it has this property, but actually why cannot I consider the same curve but with a different parametrisation to be geodesic as well?

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For definiteness, let $(M, g)$ be a connected Riemannian manifold. Generally (when pressed for detail), geometers often define:

  • A (regular) path in $M$ to be a smooth map $\sigma:I \to M$ whose domain $I$ is a non-empty interval of real numbers, and whose velocity is never $0$.

  • A (smooth) curve to be the image of a regular path. (It's not uncommon to assume in addition that $\sigma$ is a homeomorphism onto its image.)

  • A geodesic to be a critical point of the energy functional $$ E(\sigma) = \int_{a}^{b} \|\sigma'(t)\|^{2}\, dt $$ for all $[a, b] \subset I$. (This corresponds to a solution of the geodesic equation, and happens to be a critical point for the arc length functional on every closed, bounded subinterval of $I$. Volume II of Spivak's Comprehensive Introduction to Differential Geometry has a detailed discussion, if memory serves. See also geodesic computation: "energy" minimization versus arc length minimization and Critical Curves of the Energy Functional are Geodesics.)

  • A pregeodesic to be the image of a geodesic.

Informally, by contrast, the terms "path" and "curve" are often used interchangeably, and "pregeodesics" are often called "geodesics".

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