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Suppose $(M,g)$ a clsoed Riemannian manifold and $\nabla$ its Levi-Civita connection and $f$ a smooth function. Is there a known notation for $\mathsf{tr}\ \nabla_{\nabla_{\bullet} \bullet}f=\sum_i \nabla_{\nabla_ii}f$?

By simplifying it using definition of Christoffel symbols: $$\sum_i \nabla_{\nabla_ii}f=\sum_i \nabla_{\Gamma_{ii}^k\partial_k}f=\sum_i \Gamma_{ii}^k\nabla_{\partial_k}f=\sum_i \Gamma_{ii}^k {\partial_k}f$$

and the question reduce to understand that what is the $\sum_i \Gamma_{ii}^k$ but this remind me nothing. I have googled "trace of Christoffel symbols" but the search result was this MSE post that deals with $\sum_k \Gamma_{ik}^k$ instead of $\sum_i \Gamma_{ii}^k$.

Motivation: This equation comes from computing the trace of hessian of $f:=\frac{1}{2}\|X\|^2$ for some vector field $X$.

$\small{\text{mini-question:}}$ As I know, here we cannot use normal coordinates technique because the wanted quantity is not tensorial. Am I right?

C.F.G
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  • How did you get to the point where you want to use the symbol $\nabla_i i$ to denote $\sum_{k}\Gamma^{k}{ii}\partial_k$? Also, where did you find the expression $\sum{ik}\Gamma^k_{ii}\partial_k f$? If you just came up with it, I doubt there is a stablished notation for it. – Jackozee Hakkiuz May 15 '21 at 06:46
  • @JackozeeHakkiuz: it is not $\nabla_ii$ instead $\nabla_{\nabla_ii}$ but their action on functions are equal but not action on tensors. Also It appeared in the trace of hessian of $f$. – C.F.G May 15 '21 at 06:59
  • The trace of the Hessian is $$\mathrm{tr}(Hf) = \sum_{\alpha,\beta}g^{\alpha\beta}\left(\partial_\alpha\partial_\beta f - \sum_\gamma\Gamma^\gamma_{\alpha\beta}\partial_\gamma f\right)$$ I guess you mean the term $\sum_{\alpha,\beta,\gamma}g^{\alpha\beta}\Gamma^\gamma_{\alpha\beta}\partial_\gamma f$. Thanks for answering. Still, I don't see why you would write it like that. Do you have any reference of usage of that notation? This is to satisfy my own curiosity, and I guess for the clarity of the question. – Jackozee Hakkiuz May 15 '21 at 07:13
  • @JackozeeHakkiuz: more specifically I am computing the hessian of $f:=\frac{1}{2}|X|^2$ then tracing the result. – C.F.G May 15 '21 at 07:23
  • So... why do you want to find an alternative notation for $\sum_{\alpha,\beta,\gamma}g^{\alpha\beta}\Gamma^\gamma_{\alpha\beta}\partial_\gamma f$? Personally, I think it is pretty clear as it is written.

    I don't see how the form of the $f$ you are working with has any impact on the notation.

     – Jackozee Hakkiuz May 15 '21 at 07:25
  • @JackozeeHakkiuz: You meant there is no other known symbol for this quantity like Laplacian or something similar? – C.F.G May 15 '21 at 07:27
  • There is none that I personally know of. I'm not saying no such notation exists, I'm just saying that I don't see the need for such a notation. – Jackozee Hakkiuz May 15 '21 at 07:33

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