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 <i>increase</i>, whereas for a timelike arc in a Lorentzian manifold, it will be a <i>decrease</i>. 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?<br /> <br /> (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! <i>Globally</i> there may very well be more than one geodesic arc between P,Q, with different lengths.)[/itex]

 

[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.