High order derivatives on manifold

Question

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?

Answer 1

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 $$ \begin{align} (\nabla \nabla f) (X, \varphi Y) & = \nabla_X \big( \nabla_{\varphi Y} f \big) - \nabla_{\nabla_X(\varphi Y)} f \\ & = (\nabla_X \varphi) \nabla_Y f + \varphi \nabla_X \nabla_Y f - \nabla_{(\nabla_X \varphi)Y + \varphi \nabla_X Y} f \\ & = \varphi \nabla_X \nabla_Y f + (\nabla_X \varphi) \nabla_Y f - \nabla_{(\nabla_X \varphi)Y} f - \nabla_{\varphi \nabla_X Y} f \\ & = \varphi \nabla_X \nabla_Y f + (\nabla_X \varphi) \nabla_Y f - (\nabla_X \varphi)\nabla_{Y} f - \varphi \nabla_{\nabla_X Y} f \\ & = \varphi \nabla_X \nabla_Y f - \varphi \nabla_{\nabla_X Y} f = \varphi (\nabla \nabla f) (X, Y) \end{align} $$