Indices at the left of a tensor in mathematical physics/differential geometry?

Question

I am a mathematician and I am reading a paper in mathematical physics and I found the following notation:

Let $Y$ be a two–form on $M$ such that $$\nabla({}_iY_j)_k = 0.$$

Here, $\nabla$ is supposed to be a connection. I have never seen that notation in mathematics (lower-left indices)? Could someone help me understand it?

Answer 1

Here, $i$ is, if you like, the index corresponding to the direction of the covariant derivative. More precisely, for, say, a $1$-form $\eta$, $\nabla_i \eta_j$ is the abstract index notation for denotes the $2$-tensor $\nabla \eta$ defined by $$(\nabla \eta)(X, Y) := (\nabla_X \eta)(Y).$$ Indeed, in abstract index notation this quantity is $\nabla_i \eta_j X^i Y^j$.

Perhaps you're more familiar with the notation $\eta_{j, i}$, where $_,$ denotes the covariant derivative. (This is quite compact, but it is sometimes less convenient, especially when more than connection is in play.)