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

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 $$\nabla_{X_i} X_j = \sum_k \Gamma_{ij}^k X_k$$?

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.