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In Euclidean space, there can be several definitions that makes a straght line:

1. Line of shortest distance between two points
2. Line that is linear, i.e with the points satisfying a linear equation
3. Line of zero curvature everywhere
4. Line with constant curvature that is equal from both sides

The Wikipedia article states:

In geometry, it is frequently the case that the concept of line is taken as a primitive.

When one enters non-Euclidean geometry, are the above definitions consistent with each other? Definition 1, for example, becomes a geodesic.

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2 Answers

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None of these definitions as satisfactory as they rely on other concepts, such as distance or curvature, even less defined.

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A line is straight(in plane or curved space) if the covariant derivative of the velocity curve along the tangent space of it is $0$, i.e. $\nabla_\dot\gamma\dot\gamma=0$. It can be thought as a geodesic.

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