Can a geodesic always be extended?

A geodesic is like the shortest path between two points on a surface, but sometimes that path can't go on forever.

Imagine you're walking on a flat playground. No matter how far you walk, you can just keep going straight, your path can always be extended. That's easy! But now imagine you're walking on a sphere, like the Earth. If you start at the North Pole and walk straight toward the equator, eventually you'll reach the equator, but if you try to keep going straight, you’ll end up walking in a circle around the world.

So while your path can be extended for a while, it might not go on forever, just like how a rubber band stretches until it snaps. A geodesic is like that rubber band, sometimes it can stretch and keep going, but other times it hits a limit or loops back on itself.

Some surfaces are "nice" and let you walk forever, like a flat floor. Others, like the Earth, make your path curve and loop. If the surface is curved enough, your geodesic might not be able to go on infinitely, it could run into an edge or just keep going in circles.

So yes, sometimes a geodesic can be extended, but not always! A geodesic is like the shortest path between two points on a surface, but sometimes that path can't go on forever.

Imagine you're walking on a flat playground. No matter how far you walk, you can just keep going straight, your path can always be extended. That's easy! But now imagine you're walking on a sphere, like the Earth. If you start at the North Pole and walk straight toward the equator, eventually you'll reach the equator, but if you try to keep going straight, you’ll end up walking in a circle around the world.

So while your path can be extended for a while, it might not go on forever, just like how a rubber band stretches until it snaps. A geodesic is like that rubber band, sometimes it can stretch and keep going, but other times it hits a limit or loops back on itself.