How do migratory birds fly in geodesics? Recent studies point to cytochrome molecules in the eyes of migratory birds as a key component for birds to use when flying across the globe. It is thought they help the birds navigate using the earth’s magnetic field. This theory involves quantum mechanics.


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*Are there other explanations for how birds can stay in a geodesic path when migrating – do any involve birds sensing the intrinsic curvature of the earth’s surface?


It seems all that is needed is a constant direction, with a stable altitude. Yet more is needed : a latitude line, based on direction, is not a geodesic. It seems unlikely that a bird could determine a quantity such as spherical curvature.
 A: Imagine you were standing on the equator pointed west. What would happen if you walked straight ahead? You would follow the equator. You would follow a geodesic by walking in a straight line.
Geodesics on the Earth are great circles, like the equator and lines of longitude. Anything a bird can do the keep going in a straight line will work. A bird might use magnetic navigation, or use the sun or stars.
On the other hand, birds may follow coastlines, mountain ranges, or convenient winds to get where they are going. That is, them may not follow a geodesic. I don't think all the mechanisms are known.

Update after seeing the link Do birds ever fly in clouds? from the comments below.
I do not believe birds fly inside clouds on purpose. If they do, I do not expect they fly in straight lines.
The link is to a post in the biology forum. An answer to the forum post has a link to a pay walled Wilson Bulletin article Spatial Disorientation in Birds . The answer quotes from the article

The only conclusion is that birds are susceptible and suffer from spatial disorientation, and further that the causes of spatial disorientation in birds are exactly the same as those which affect the human pilot, namely; (a) the loss of true visual cues to the horizontal; (b) inexperience in flying under such conditions where visual cues are lost;

This is reasonable.
The answer has another link from Science, Frigate birds track atmospheric conditions over months-long transoceanic flights.
The abstract says

Understanding how animals respond to atmospheric conditions across space is critical for understanding the evolution of flight strategies and long-distance migrations. We studied the three-dimensional movements and energetics of great frigate birds (Fregata minor) and showed that they can stay aloft for months during transoceanic flights. To do this, birds track the edge of the doldrums to take advantage of favorable winds and strong convection. Locally, they use a roller-coaster flight, relying on thermals and wind to soar within a 50- to 600-meter altitude band under cumulus clouds and then glide over kilometers at low energy costs. To deal with the local scarcity of clouds and gain longer gliding distances, birds regularly soar inside cumulus clouds to use their strong updraft, and they can reach altitudes of 4000 meters, where freezing conditions occur.

This says that birds fly inside clouds on purpose to take advantage of the energy saving updrafts. On closer inspection, I think the article is likely mistaken on this point. It is more likely that birds fly in the strong lift below clouds, following the same kinds of behavior that hang glider pilots do. This doesn't change any conclusions from the rest of the article.
The article describes how birds were equipped with data loggers. It says

No information was available to us on the presence of cumulus clouds on the tracks. On
figures 2 or 4 of the presence of clouds was derived from the behaviour of birds. An
ascending flight without or with little flapping of wings can only be made with support of an
ascending air current. At sea an ascending air current can only exist in a thermal below a
cumulus cloud. In an area not below a cumulus cloud, there is no rising air. In this area the air
is moving only horizontally or moving downward in terms of a downdraft. Thus, it can be
concluded that there is a cumulus cloud where the bird performs an ascending flight, when
birds climb without flapping their wings. Soaring outside but close to the clouds would not be
possible because of the downdrafts occurring in the near vicinity of cloud-edges (Fig. 4).

Hang gliders often fly in mountains, where a series of cumulus clouds along a range form a "cloud street". They climb under a cloud, glide to the next, and climb again.
Hang glider pilots avoid clouds getting sucked into clouds because

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*The currents in or near large clouds are so strong you can't control the hang glider. In fact, the hang glider might break.

*You can get electrocuted from lightning

*You can get sucked up to altitudes where you can't breathe and/or you would freeze.

*You can't see where you are going.

The article goes on to show information about the flights of various birds. The $4000$ m flight was a single outlier. Here is the altitude data for that $3$ day flight over the open ocean.

The graph shows a mix of high and low altitude climb-and-glide flight. The peak climb was far higher than this or any other bird made. Note how sharp the peak is. The bird did not spend much time above $3000$ m, and therefore did not glide very far in that time. Note how many high altitude climbs the bird made that day. The weather may have been more cloudy that day, with more strong updrafts.
Here is a chart of max altitude and ascent rate for $263$ climbs by $8$ birds.

Note this climb had the $8^{th}$ highest ascent rate, perhaps $5$ times the typical rate.
My personal theory is that the bird was sucked up into a cloud and could not escape until reaching that altitude. When she did escape, perhaps she was in a strong downdraft or perhaps she was frightened. Either way, the first $1000$ m is a very rapid descent. She did not climb to high altitude for the rest of the voyage.
It might be that all the most rapid ascents were similar, but did not go up so far. However, there are a few ascents above $2500$ m at rates that look much more voluntary. So it is possible that all these high flights were purposeful.
