1. parametrized curve

I am taking a Differential Geometry class, and we started talking about parametrized curve. There is something I don't understand. A parametrized curve alpha(t) has the property that its second derivative is identically zero, then what can be said about alpha(t)? Can someone help?

2. In mechanics, one would say: if the acceleration (second derivative) is zero, then the speed is constant, and therefore the point moves along a line, uniformly.

In other words, $\displaystyle \alpha(t)=\alpha(0)+t\vec{v}$ for some vector $\displaystyle \vec{v}$.

3. Originally Posted by dori1123
I am taking a Differential Geometry class, and we started talking about parametrized curve. There is something I don't understand. A parametrized curve alpha(t) has the property that its second derivative is identically zero, then what can be said about alpha(t)? Can someone help?
its a geodesic, this means, if you have two points $\displaystyle A$ and $\displaystyle B$ on the surface $\displaystyle M$, and $\displaystyle \alpha$ passes trought this points, then the shortest way on $\displaystyle M$ to travel from $\displaystyle A$ to $\displaystyle B$ is traveling trough the curve $\displaystyle \alpha$, provided that its a georesic, i.e. $\displaystyle \alpha^{\prime\prime}\equiv0$.

Note. Recall that on a plain, the shortest way to reach from $\displaystyle A$ to $\displaystyle B$ is the way combined combined by a line, which is obtained by solving the differential equation $\displaystyle \alpha^{\prime\prime}(t)=0\text{ for }t\in\mathbb{R}$.

4. It depends indeed on what is meant by $\displaystyle \alpha''(t)$ in your Geometry class, dori1123.

If it is just the usual derivative in $\displaystyle \mathbb{R}^3$, then my first answer suffices.

It may however also denote the projection of $\displaystyle \alpha''(t)$ (defined in $\displaystyle \mathbb{R}^3$ the usual way) on the tangent plane of $\displaystyle \mathcal{M}$ at $\displaystyle \alpha(t)$. In this case, this is an "intrinsic way" (independent of the embedding of $\displaystyle \mathcal{M}$ in $\displaystyle \mathbb{R}^3$) to define the second derivative of the parametrized curve. It can be defined in an more general context using affine connections and is then denoted $\displaystyle \nabla_{\alpha'(t)}\alpha'(t)$. And $\displaystyle \nabla_{\alpha'(t)}\alpha'(t)=0$ is the equation of the geodesics, as bkarpuz writes.

On a surface in $\displaystyle \mathbb{R}^3$, geodesics are characterized by the fact that their acceleration lies in the orthogonal of the tangent plane: $\displaystyle \alpha''(t)\in T_{\alpha(t)}^\perp$. In other words, for geodesics, the projection of $\displaystyle \alpha''(t)$ on the tangent plane at $\displaystyle \alpha(t)$ is $\displaystyle 0$.