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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!

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    yes - I did a new question at : http://math.stackexchange.com/questions/261850/commutators-and-christoffel-symbols-in-a-non-holonomic-basis If you want, I can validate your reply that gives the wiki link: I guess this is a major change that might make things work!2012-12-19

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As the discussion in the comments revealed, the major problem that may affect the calculations is that in a frame is not arising from a coordinate system, so called non-holonomic frame, the Christoffel symbols of the Levi-Civita connection of the metric $g$ are given by a formula that is slightly different. See in Wikipedia here.

This formula may be obtained by a generalization of the standard calculation of the Christoffel symbols of the Levi-Civita connection that you can find here, taking into account that the Lie brackets of the elements of the non-holonomic frame need not commute: $ [\vec{e}_i,\vec{e}_j] := \nabla_{\vec{e}_i}{\vec{e}_j} - \nabla_{\vec{e}_j}{\vec{e}_i} = c_{ij}{}^k \vec{e}_k $

Remark. The elements of such a frame should not be denoted as $\partial_i$ but rahter as $\vec{e}_i$, $\mathbf{e}_i$ or $\mathbf{u}_i$ etc. Using $\partial_i$ is conventionally reserved for vectors form a coordinate frame.