Straight lines and flat surfaces

[birulami]

Suppose I have a parameterized line $\phi:\mathbb{R}\to\mathbb{R}^n$ given by $\phi(t) = (x^\mu(t))|_{\mu=1}^n$. How can I tell that the line is straight.

My best answer so far is that at every time $t$ the acceleration (2nd derivative) is parallel to the velocity (1st derivative), i.e. $\ddot{\phi}(t) = g(t)\cdot\dot{\phi}(t)$, for some function $g(t)$ (which likely should better not be zero anywhere).

Is this a valid description (necessary and sufficient) of a straight line? Are there different ones?

And a very similar question for a 2-dimensional surface, i.e. now we have $\phi:\mathbb{R}^2\to\mathbb{R}^n$. Assuming the above is true for the straight line description. Is there a similar condition for the surface?

 

[CompuChip]

Since you have posted this in the Differential Geometry forum, this may be a good time to read (up) on geodesics which is the generalization of the idea of "straight line".

 

[HallsofIvy]

If you really mean "straight" rather than "geodesic", a line is straight, at a given point, if and only if its second derivative, with respect to arc length, is 0.

 

[lavinia]

.

My best answer so far is that at every time $t$ the acceleration (2nd derivative) is parallel to the velocity (1st derivative), i.e. $\ddot{\phi}(t) = g(t)\cdot\dot{\phi}(t)$, for some function $g(t)$ (which likely should better not be zero anywhere).

And a very similar question for a 2-dimensional surface, i.e. now we have $\phi:\mathbb{R}^2\to\mathbb{R}^n$. Assuming the above is true for the straight line description. Is there a similar condition for the surface?

This is correct. A straight line can be parameterized to have the equation c(t) = b + at for vectors a and b. Any path that follows this line will be a reparameterization of it i.e. t = f(s). You can prove your conclusion using the Chain Rule.

The same idea applies for the map of a plane into Euclidean space or for a map of R^m into R^n. The only difference is the number of parameters.

 

[blue_raver22]

Your answer is a valid description for a straight line, but it is not necessary and sufficient. A more precise definition of a straight line is that it is the shortest path between two points in a given space. In other words, if you consider any two points on the line, the distance between them along the line is always shorter than any other path between those two points. This can also be expressed mathematically as the line having constant curvature and torsion, which means that it does not bend or twist in any way.

For a 2-dimensional surface, a similar condition would be that it has constant Gaussian curvature, meaning that the curvature at any point on the surface is the same in all directions. This would result in a surface that is flat and does not curve in any direction. However, this condition is not necessary and sufficient for a flat surface, as there are other surfaces that can have constant Gaussian curvature but are not flat.

Ultimately, the concept of straight lines and flat surfaces is dependent on the geometry of the space in which they are defined. In Euclidean geometry, the definitions above hold true, but in other geometries such as non-Euclidean or curved spaces, the definitions may differ.