Some conditions required of a covariant derivative operator

Question

The following content about Differential Geometry comes from the General Relativity book written by Robert M. Wald. Compared with ordinary DG books, its material may not be in the mainstream, but I have no choice but to figure it out because the only access to GR for me is via this book.

A tensor of type $(k,\ell)$ over a real vector space $V$ is defined to be a multi-linear map from $(V^*)^k\times V^\ell$ to $\mathbb R$, where $V^*$ is the dual space of $V$. Of all the underlying spaces $V$, the most important case is the tangent space $T_p M$ to a manifold $M$ at $p\in M$. And an assignment of a tensor over $T_p M$ to each $p\in M$ is called a tensor field. Finally, a covariant derivative operator $\nabla$ on $M$ is a mapping that takes each tensor field of type $(k,\ell)$ to a tensor field of type $(k,\ell+1)$ and satisfies the five conditions listed below.

1. Linearity
2. Leibnitz rule
3. Commutativity with the contraction operation
4. Consistency with the notion of tangent vectors as directional derivatives on scalar fields
5. Torsion free

I'd like to ask about the fifth condition: torsion free. According to the author, this means that for each function $f$ on $M$, we have $$\nabla_a\nabla_b f=\nabla_b\nabla_a f.\tag{*}$$

But how can the operator $\nabla$ act on a scalar field $f$? Also, I cannot understand the meaning of the expression (*). What are the lower indices $a,b$ for? Can anyone help me dig into this? Thank you so much.

Answer 1

Smooth functions are $(0,0)$-tensor fields, so the covariant derivative has to act on them. As mentioned in the comment by @SiKucing, condition 4. actually refers to the action of $\nabla$ on $(0,0)$-tensor fields. If you are familiar with this, it says that on functions you get the ordinary (exterior) derivative $df$. One way to phrase the torsion-freeness condition is that the $(2,0)$-tensor field $\nabla df$, which in each point takes as input two tangent vectors and produces a number, is symmetric in the two vectors. This is what the index expression says (regardless of whehter these are abstract indices or coordinate indices).