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Let $M$ be a smooth manifold and $\nabla$ the Levi-Civita connection. Now, I am a bit puzzled by a serious notational ambiguity, namely for the second covariant derivative.

To explain myself, let us consider a vector field $v\in\Gamma(TM)$. Then, the notation $\nabla_{a}\nabla_{b}$ can mean two things:

  1. Some authors use the notation $\nabla_{a}:=\nabla_{\partial_{a}}$. In this case, $\nabla_{\partial_{b}}v\in\Gamma(TM)$ and $\nabla_{\partial_{a}}(\nabla_{\partial_{b}}v)\in\Gamma(TM)$. In local coordinates, we get $$\nabla_{\partial_{a}}\nabla_{\partial_{b}}v=(\partial_{a}(\partial_{b}v^{c}+\Gamma_{db}^{c}v^{d})+\Gamma_{ad}^{c}(\partial_{b}v^{d}+\Gamma_{eb}^{d}v^{e}))\partial_{c}$$
  2. Some authors, especially in the physics literature (but for example also in Wald's book on GR), write expressions like $\nabla_{a}\nabla_{b}v^{c}$ to indicate the coefficients of the $(1,2)$ tensor obtained by applying $\nabla$ twice. In other words, $\nabla_{a}$ is acting on the $(1,1)$ tensor with coefficients $\nabla_{b}v^{c}=\partial_{b}v^{c}-\Gamma_{bd}^{c}v^{d}$, i.e. $$\nabla_{a}\nabla_{b}v^{c}=\partial_{a}(\partial_{b}v^{c}+\Gamma_{db}^{c}v^{d})+\Gamma_{ad}^{c}(\partial_{b}v^{d}+\Gamma_{eb}^{d}v^{e})\color{red}{-\Gamma_{ab}^{d}(\partial_{d}v^{c}+\Gamma_{de}^{c}v^{e})}\quad (=:v^{c}_{;a;b})$$

So, the difference is basically the red term. This is actually well-known: Usually, one defines the second covariant derivative globally as the operator $$\nabla^{2}_{X,Y}v:=\nabla_{X}\nabla_{Y}v\color{red}{-\nabla_{\nabla_{X}Y}v}$$ where the second piece essentially corresponds to the red stuff appearing in 2. In other words, the construction in (1.) is the piece $\nabla_{\partial_{a}}\nabla_{\partial_{b}}v$, while the construction in (2.) corresponds to the full second covariant derivative $\nabla_{\partial_{a},\partial_{b}}v$.

Now, my problem is, since some authors unfortunately use the notation $\nabla_{a}\nabla_{b}$ for $\nabla_{\partial_{a}}\nabla_{\partial_{b}}$, how can I know which one of the above someone is using? As an explicit example: Consider a Riemannian manifold $(M,g)$. Then, one can find in many books that the linearization of the Ricci tensor takes the form $$\mathrm{Ric}(g)_{ab}\xrightarrow{g=g_{0}+\lambda h+\mathcal{O}(\lambda^{2})}-\frac{1}{2}g_{0}^{cd}\nabla_{c}\nabla_{d} h_{ab}+\frac{1}{2}(2\nabla_{c}\nabla_{(a}h^{c}_{b)}-\nabla_{a}\nabla_{b}\mathrm{tr}_{g_{0}}(h))$$ where all the covariant derivatives are with respect to the background metric $g_{0}$. Now, in this formula, is $g_{0}^{cd}\nabla_{c}\nabla_{d}$ as in my point (2.) above (=the connection Laplacian) or as in my point (1.), i.e. $g_{0}^{cd}\nabla_{\partial_{c}}\nabla_{\partial_{d}}$?

I know this is kind of a vague question, but this a point that really puzzles me. I am just wondering whether there is a general way to tell which convention is used in some specific text (without going through the full calculation, like for example deriving the linearization of the Ricci tensor myself).

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1 Answer 1

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One style is in index-free notation and the other is in index notation. You know which style you're looking at, based on whether vectors and tensors are written with indices all the time.

  1. In index-free notation, vectors and tensors are written like $v$, $h$, etc., and covariant derivatives are written like $\nabla v$ or $\nabla_X h$.

  2. In index notation, vectors are written like $v^a$, $h_{ab}$, etc., and covariant derivatives are written like $\nabla_a v^b$ or $X^a \nabla_a h_{bc}$. (Or $v^b{}_{;a}$ or $h_{bc;a} X^a$, but we'll set that aside.) In this system, the meaning of $\nabla_a \nabla_b v^c$ or $\nabla_c \nabla_d h_{ab}$ is always (2).

    The index-free notation $\nabla^2_{X,Y} v$ translates to $X^a Y^b \nabla_a \nabla_b v^c$, while $\nabla_X \nabla_Y v$ translates to $X^a \nabla_a (Y^b \nabla_b v^c)$. Their difference is $X^a (\nabla_a Y^b) (\nabla_b v^c)$ which translates back to $\nabla_{\nabla_X Y} v$.

Even if $\nabla_a v$ could be short for $\nabla_{\partial_a} v$, that's still a form of system #1, as evidenced by the lack of an index on "$v$". In system #2, the index on "$v$" in $\nabla_a \nabla_b v^c$ is mandatory; $\nabla_a \nabla_b v$ is invalid in system #2.

Other incidental features of the different traditions can be distinctive too. For example, the use of parentheses $T_{(ab)}$ as notation for $\frac{1}{2} (T_{ab} + T_{ba})$, e.g. in $\nabla_c \nabla_{(a} h^c{}_{b)}$, is characteristic of style #2.

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