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?
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, for some vector .
It depends indeed on what is meant by in your Geometry class, dori1123.
If it is just the usual derivative in , then my first answer suffices.
It may however also denote the projection of (defined in the usual way) on the tangent plane of at . In this case, this is an "intrinsic way" (independent of the embedding of in ) 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 . And is the equation of the geodesics, as bkarpuz writes.
On a surface in , geodesics are characterized by the fact that their acceleration lies in the orthogonal of the tangent plane: . In other words, for geodesics, the projection of on the tangent plane at is .