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Suppose I have a Riemannian manifold $M$ and a smooth function $f:M \to \mathbb{R}$. I denote the gradient of $f$ with $\nabla f$. What is the meaning of $$ \nabla^N f$$ with $N \ge 3$ integer? Is it the $N$ covariant tensor field $$ \nabla ^N f (X_1, \dots, X_N ) = \langle \nabla_{X_1} \nabla_{X_2} \dots \nabla_{X_{n-1}} \nabla f , X_N \rangle $$ ? In any case, what is the meaning of $$ \int_M \nabla^N f \text{dm} $$ with m the volume measure?

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    $\begingroup$ In general, $\nabla^{k+1}f(X_0, X_1, ...,X_k)=\nabla_{X_0}(\nabla^kf)(X_1,..., X_k)$. What is the context of your integral? Maybe $\nabla^N$ stands for some normal derivative? $\endgroup$ – Yu Ding Mar 30 at 6:51
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An explicit expression for $\nabla^{k+1}f(X_0, X_1, ...,X_k)=\nabla_{X_0}(\nabla^kf)(X_1,..., X_k)$ from a comment to the question can be given as $$ \begin{align} (\nabla^{N+1} f) (X_0, X_1, \dots, X_n) = & \nabla_{X_0} \big( (\nabla^{N} f) ( X_1, \dots, X_n) \big) \\ & - (\nabla^{N} f) (\nabla_{X_0} X_1, \dots, X_n) \\ & \dots \\ & - (\nabla^{N} f) (X_1, \dots, \nabla_{X_0} X_n) \end{align} \tag{1} $$ (see Wikipedia).

This is because we want all $f$, $\nabla f$, $\nabla \nabla f$, and so on, to be tensors, so that $\nabla^N f$ is a $N$-covariant tensor, that is a multilinear function of all its variables ($X_0$, ..., $X_{N-1}$). In other words, we want $\nabla^N f$ to be linear at each slot.

To see why we need to subtract all those things in the lower lines in (1), let us do a simple calculation: $$ \nabla_X\big(\nabla_{(\varphi Y))}f \big) = \nabla_X (\varphi \nabla_Y f) = (\nabla_X \varphi) \nabla_Y f + \varphi \nabla_X \nabla_Y f $$ which shows that $\nabla_X (\nabla_Y f)$ is nonlinear in slot $Y$, whereas $$ \begin{align} (\nabla \nabla f) (X, \varphi Y) & = \nabla_X \big( \nabla_{\varphi Y} f \big) - \nabla_{\nabla_X(\varphi Y)} f \\ & = (\nabla_X \varphi) \nabla_Y f + \varphi \nabla_X \nabla_Y f - \nabla_{(\nabla_X \varphi)Y + \varphi \nabla_X Y} f \\ & = \varphi \nabla_X \nabla_Y f + (\nabla_X \varphi) \nabla_Y f - \nabla_{(\nabla_X \varphi)Y} f - \nabla_{\varphi \nabla_X Y} f \\ & = \varphi \nabla_X \nabla_Y f + (\nabla_X \varphi) \nabla_Y f - (\nabla_X \varphi)\nabla_{Y} f - \varphi \nabla_{\nabla_X Y} f \\ & = \varphi \nabla_X \nabla_Y f - \varphi \nabla_{\nabla_X Y} f = \varphi (\nabla \nabla f) (X, Y) \end{align} $$

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  • $\begingroup$ Thank you very much. What about the integral of a map $x \to T_y M$ over $M$ for some fixed $y \in M$? $\endgroup$ – Bremen000 Apr 1 at 17:03
  • $\begingroup$ @Bremen000 Your questions regarding integration are not very clear to me. I agree with the comment above that what you've mentioned in the question looks like an integral of a normal derivative. In general, it is not clear what an integral of a section of a vector bundle means, because the construction of the integral assumes special quantities to be integrated. See this answer for more insights, or create please a separate question. $\endgroup$ – Yuri Vyatkin Apr 1 at 21:31
  • $\begingroup$ I have already seen the answer you cite. My problem is, I think, simpler. In that answer they were asking about the integral of a vector field whereas I'm just considering a map from the manifold to a fixed tangent space, not all the vector bundle. In my question it was (not even barely) clear. I have edited it. $\endgroup$ – Bremen000 Apr 2 at 6:20
  • $\begingroup$ @Bremen000 You should not have edited your question to that degree, because it's basically become a completely different question, and the meaning of the comment is lost. It would much more beneficial for all of us if you restored the original question and created a new question, the link to which you could provide here in the comments. Otherwise our collaboration with the local community is going to be not very productive, I am afraid... $\endgroup$ – Yuri Vyatkin Apr 2 at 10:19
  • $\begingroup$ You are right. I did as you suggested, this is the link to the new question. $\endgroup$ – Bremen000 Apr 2 at 12:03

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