Geodesics through parametrization I have the following general question about geodesics.
I know the following equation for a geodesic $\sigma$ on a manifold $M\subset R^n$ of dimension $m$, written in local coordinates:
$${\sigma^k}^{''} (t) + \Gamma_{i,j}^k {\sigma^{i}}'{\sigma^{j}}'=0,$$
for $i,j,k=1, \dots, m$.
Now, if I have a curve $\gamma(t)=(\gamma_1(t), \dots, \gamma_n(t))$ in $M$, how can I check that such a curve is a geodesic?
More precisely, how can I write my curve in local coordinates, in order to check if it satisfies my equation? I am stuck. Examples are really welcomed too.
Thank you.
 A: With a parametrization $\Phi$ from an open set $U$ of $\Bbb R^m$ and a curve $\sigma$ in $U$ you will get $\gamma=\Phi\circ\sigma$. For that $\sigma$ you ought to get your $\sigma^k$ in the geodesic equations.
A: If you have your $M$ to be an $m$-dimensional submanifold of $\mathbb{R}^n$ and the Riemannian metric on $M$ is the Euclidean metric of  $\mathbb{R}^n$ restricted to $M$, then there are two things you need to check.
1) Make sure the curve $\gamma : (a,b) \to  \mathbb{R}^n$ lies on $M$, i.e. $\gamma(t) \in M$ for all $t \in (a,b)$. 
2) Compute the normal curvature vector $N(t)$ of $\gamma(t)$ (the curvature vector of $\gamma(t)$ which is orthogonal to the tangent of $\gamma(t)$ and points to the pivoting point around which $\gamma$ is turning the most at time $t$) and check that $N(t)$ is perpendicular to the tangent space $T_{\gamma(t)}M$ for all $t \in (a,b)$.
If $t$ is an arclength parametrization of $\gamma(t)$, which is equivalent to $\big(\dot{\gamma}(t) \cdot \dot{\gamma}(t) \big)\equiv 1$, then $N(t) = \ddot{\gamma}(t)$.    
