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?


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".


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.