I am (numerically) computing the second fundamental form for a curve $\gamma(t)$ embedded in a Riemannian manifold $(M, g)$. I would like to double check if what I am doing is correct.
First, define a basis where the first element of the basis is $\dot\gamma(t)$ (which I numerically evaluate) and the other elements are arbitrary (not necessarily orthogonal) fixed vectors in $TM$ (we'll suppose all the vectors span $TM$ and are thus independant). My basis vectors are called $\partial_i$.
Compute the Levi-Civita connection $\nabla_{\dot\gamma}\dot\gamma$ using the Christoffel symbols $\Gamma^i_{j,k}$. In particular, I use the standard formula (that uses $\frac{\partial g_{i,j}}{\partial_k}$ etc.) with no particular change due to the non orthogonality of the basis. My final value is $\nabla_{\dot\gamma}\dot\gamma = \sum_i \Gamma^i_{1,1}\partial_i$
I project $\nabla_{\dot\gamma}\dot\gamma$ on the normal space $NM$ by computing $\nabla_{\dot\gamma}\dot\gamma - \frac{g(\nabla_{\dot\gamma}\dot\gamma,\dot\gamma)}{g(\dot\gamma,\dot\gamma)}\dot\gamma$
In particular, my second step does not seem to lead to $\nabla_{\dot\gamma}\dot\gamma=0$ for geodesics, which could be either due to my reasoning or other factors (bugs, numerical approximations etc.). Is-there a problem with the method above ? Am-I allowed to use a non orthogonal basis ? My second doubt is about $\nabla_{\dot\gamma}\dot\gamma = \sum_i \Gamma^i_{1,1}\partial_i$ : is it correct ?
Thanks!