# Definition A *curve* is a [[Differentiable Function|differentiable]] [[Types of Maps|injection]] from an [[open set]] of $\mathbb{R}$ [[Types of Maps|into]] a [[manifold]] $M$. More precisely, if the curve is defined on the interval $I \subset \mathbb{R}$, then $\gamma: I \rightarrow M$ is a $C^k$-curve if the component functions of $\gamma$ are $C^k$ differentiable. One associates with each point $\lambda \in I$ a point in $M$, called the *image point* of $\lambda$. More broadly, the *image* of the curve is $\gamma[I] \subseteq M$. This curve is parameterized by $\lambda$ and the image $\gamma[I]$ gives us the ordinary notion of a curve. Two curves are different even if they have the same image in $M$, because they can are in principle parameterized differently, i.e. they assign different parameter values to each image point. ^6a34c8 ![[Pasted image 20210123191354.png]]