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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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    Vector $\nabla_{\dot\gamma}\dot\gamma - \frac{g(\nabla_{\dot\gamma}\dot\gamma,\dot\gamma)}{g(\dot\gamma,\dot\gamma)}\dot\gamma$ lives not in $NM$ (which is not defined here!) but in the normal bundle along $\gamma$ (which is spanned by the remaining vectors from your frame: $N\gamma = \operatorname{span}(\partial_2,\dots,\partial_n)$)2012-12-16
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    yep, that's what I meant by $NM$ ($N$ for normal in "normal bundle" like $T$ in $TM$ for "tangent bundle of M") - isn't it standard notation ? thanks!2012-12-17
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    Usually $NM$ denotes the normal bundle of an immersed submanifold $M$, so in your case something like $N \gamma$ looks more appropriate, because your $\gamma$ is a submanifold in $M$ in a certain sense (if it is regular and injective).2012-12-17
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    oh yes, indeed, that makes sense - thanks2012-12-17
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    This formula $\nabla_{\dot\gamma}\dot\gamma = \sum_i \Gamma^i_{1,1}\partial_i$ is indeed correct because you should define your Christoffels this way, but dealing with a non-coordinate frame like yours may require an elevated care, because many familiar properties will not hold.2012-12-18
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    thanks - that's what I am wondering as well: I'm afraid I'm doing a silly mistake just because my basis is neither normalized not orthogonal ! Orthogonalizing it would make $\frac{\partial g_{ij}}{\partial_k}$ much more difficult to compute since all the tangent vectors would vary along the path2012-12-18
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    Oops, I just noticed that you said "I use the standard formula ... with no particular change due to the non orthogonality of the basis". Please refer to section "Connection coefficients in a non holonomic basis" [here](http://en.wikipedia.org/wiki/Christoffel_symbols). This however requires somewhat cleaner treatment that I don't have ready. May be someone can answer?2012-12-18
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    oooouch! good catch! Thanks! I'll investigate what these commutation coefficients are!2012-12-18
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    arg ; unless I have another bug, this is not enough to give me 0 for geodesics... still investigating. Thanks!2012-12-18
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    just a question: since I am dealing with matrices, the commutator is just X*Y-Y*X with standard matrix multipliation, or is it supposed to be more fancy? - I can ask a new math.stackechange question for that though, this might be more appropriate.2012-12-19
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    I am not getting clearly what your precise question is. It is more beneficial for all of us to make another MathSE question, rather then extend these comments :-)2012-12-19
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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.