Can Geodesics Be Inflectional in Euclidean Space?

[humanino]

I am trying to figure out if a geodesic can be inflectional (in euclidean space...). I am not sure it even makes sens, from the definition of a geodesic, but it seems to me that a geodesic will not in general be extremal, but only stationnary.

Is there a general theorem preventing a monster such as an "inflectional geodesics", or do you have a beautiful example, or am I just obviously on the wrong track here ?
Thank you for any help.

 

[Chris Hillman]

Rephrasing the question...

Your term "inflectional geodesic" is I nonstandard, but I think I see what you are asking. Let me rephrase it.

The stationary property says that if we consider a geodesic arc with endpoints P,Q, then making small perturbations of "size" [itex]\varepsilon[/tex] (keeping the endpoints P,Q) will change the length by only $O(\varepsilon^2)$. For an arc in a Riemannian manifold, this length change will in fact be an increase, whereas for a timelike arc in a Lorentzian manifold, it will be a decrease. So the question was: could there be some exotic signature with the property that some such perturbations of some particular geodesic arc increase the length, while others decrease it?

(As an example of local versus global distinction: as we can see by considering great circles on a globe, "small" is essential in the above! Globally there may very well be more than one geodesic arc between P,Q, with different lengths.)

 

[humanino]

could there be some exotic signature with the property that some such perturbations of some particular geodesic arc increase the length, while others decrease it?

Well, I was not thinking about something that elaborate. I am not sure that my question is equivalent (or even is implied by, or implies...) yours. So my first lesson would be to be more precise in my questions if I want an expert answer

Let me try to rephrase.

I am concerned with geodesics on surfaces embeded in euclidean spaces, so I am thinking in a physicist's manner. I should try to switch to the mathematician point of view, and think in terms of metric directly. You said the variation is $O(\varepsilon^2)$. My rephrased question could be : "Is there a general theorem preventing that the $O(\varepsilon^2)$ vanishes anywhere (between an arbitrary pair of points), whatever the metric, or could it happen that the variation is $O(\varepsilon^3)$ (what I called inflectional) ?"

 

[Chris Hillman]

Well, if your surface has sufficiently high order contact to the tangent spaces all along some geodesic arc, then sure, I suppose the variation could well be even smaller than $O(\varepsilon^2)$. I don't think that "inflectional" would be a good term at all for this kind of thing, however. And I can't understand the first alternative you tried to describe, so I guess I still don't know what the question is.