How does the solution of a differential equation on a manifold yield a map?

Question

1. In: "A solution $x^μ(λ)$ is a map from $\mathbb{R} → M$": Why is $x^μ(λ)$ considered a map and why does it go from $\mathbb{R} → M$? I can't seem to illustrate this in my mind.

2. In:"If the manifold is $\mathbb{R}^D$, then we know that the vector $dx^μ/dλ$ is tangent to the curve $x^μ(λ)$". What if the manifold is a general manifold, what happens to the tangent?

Answer 1

The most important thing is that the solution is a curve on the manifold $M$. Remember that a manifold looks locally like $\mathbb{R}^n$. Then locally you can interpret the differential equation as telling you the velocity of some point particle as a function of position in Euclidean space (think of a particle flowing through a river that has a well-defined velocity at each point). The parameter $\lambda$ is $\mathbb{R}$-valued (think of it like a time), and to each value of $\lambda$ is associated a point on the manifold $x^\mu(\lambda)$. That's all that is meant by a map from $\mathbb{R}$ to $M$.

If the manifold happens not to be $\mathbb{R}^n$, $V^\mu(x)$ is still tangent to the curve at each point - remember that locally the manifold looks like $\mathbb{R}^n$, so this is meaningful in the same sense as in Euclidean space. Actually you can probably illustrate this very well in your head - after all, we're always walking around on the surface of a sphere, a manifold that is not $\mathbb{R}^n$.