2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Oh I see. I took a course from do Carmo's book last semester, and my professor always called it del. $\endgroup$ – Aweygan Feb 3 at 19:30
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.