How to read the expression of an affine connection: $\nabla_X Y$?

Question

I am studying Riemannian Geometry from the textbook Riemannian Geometry by do Carmo (English edition). In section 2 of chapter 2, page 50, he defines an affine connection as follows:

2.1 Definition. An affine connection $\nabla$ on a differentiable manifold $M$ is a mapping $$ \nabla : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M) $$ which is denoted by $(X,Y) \overset{\nabla}{\to} \nabla_X Y$ and which satisfies the following properties:

1. $\nabla_{fX + gY} Z = f \nabla_X Z + g \nabla_Y Z$.
2. $\nabla_X(Y+Z) = \nabla_X Y + \nabla_X Z$.
3. $\nabla_X (fY) = f \nabla_X Y + X(f) Y$,

in which $X,Y,Z \in \mathfrak{X}(M)$ and $f,g \in \mathcal{D}(M)$.

Here, $M$ is a smooth manifold, $\mathfrak{X}(M)$ is the set of all smooth vector fields on $M$, and $\mathcal{D}(M)$ is the ring of real-valued smooth functions on $M$.

My question is, how do I speak (or read) the affine connection $\nabla$, or $\nabla_X Y$? do Carmo does not give any suggestions, and I was unable to find any details on Wikipedia or in Google searches with terms like

• affine connection pronounce/pronunciation
• affine connection how to speak
• affine connection how to read

I could not find any related question on this Stack Exchange either.

I am aware that the $\LaTeX$ command for $\nabla$ is \nabla. In my undergraduate classes, I encountered this symbol as a gradient operator, where we called it "grad", and also sometimes "del".

Can someone tell me what is (or what are) the standard pronunciation(s) for the affine connection $\nabla$ in Riemannian Geometry? To be precise, how do I read the symbol(s) $\nabla$ or $\nabla_X Y$ when I encounter them in the text, for instance in an expression like $\nabla_{X_i} X_j = \sum_k \Gamma_{ij}^k X_k$?

Thanks in advance.

Answer 1

One reads the symbol $\nabla$ as "del" or "nabla" and the expression $\nabla_XY$ is called the covariant derivative of $Y$ in the direction $X$. For more details see p. 89 of Introduction to Riemannian Manifolds by John M. Lee.