A curve on Riemannian Manifold is $c:I\rightarrow M$. We study many properties about it, like parallel $\bigtriangledown_\dot{c}X=0$ and geodesic $\bigtriangledown_\dot{c}\dot{c}=0$. And we apply the theories of curve on Riemannian Manifold to many theorems such as something about $\pi_1(M)$.
So I have a thought to generalize the curve on Riemannian Manifold, but not to introduce the definition of submanifold geometry. I think we can study: $r:\mathbb R^n\rightarrow M$.
It also can have meanings about "parallel" and "geodesic".
Definition A vector field $X$ along $r$ is called parallel when for any $k=1,2...,n$, $\bigtriangledown_{\frac{\partial r}{\partial x_k}}X=0$.
We have the local representation of the definition: $$\sum_{i=1}^m\sum_{j=1}^m\frac{\partial r_i}{\partial x_k}X_j\Gamma_{ij}^k+\frac{\partial X_l}{\partial x_k}=0$$Definition $r$ is called geodesic when for any $s,t=1,2,...,n$, $\bigtriangledown_{\frac{\partial r}{\partial x_s}}{\frac{\partial r}{\partial x_t}}=0$.
We also have the local representation of the definition: $$\sum_{i=1}^m\sum_{j=1}^m\frac{\partial r_i}{\partial x_s}\frac{\partial r_i}{\partial x_t}\Gamma_{ij}^k+\frac{\partial^2 r_k}{\partial x_s\partial x_t}=0$$
However, I have some confusion about my thoughts.
First, can these generalization have meanings? That is to say, are they helpful to our study? And does people study them?
Second, if they are meaningful and helpful, can someone give some examples about these?