Coincidence about nabla?

Question

I was surprised to notice that gradient of function and Levi-Civita connection have the same notation, i.e. nabla sign $\nabla$. Moreover, extending any connection on tensors, one let it be differential (or, equally in presence of Riemann metric, gradient) on functions. Is it just a strange coincidence?

Answer 1

Your observation is correct. This is not just a coincidence, but a smart choice of conventions, which allows to have the Leibniz rule written in a universal manner.

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A connection in a vector bundle $E$ over manifold $M$ is a $\mathbb{R}$-linear differential operator $D \colon \Gamma(E) \to \Gamma(T^*M \otimes E)$ between sections of vector bundles $E$ and $T^*M$, which satisfies the Leibniz rule: $$ D(f \, t) = \mathrm{d}f \otimes t + f \, Dt \tag{1} $$ for any $f \in C^{\infty}(M)$ and $t \in \Gamma(E)$.

By requiring that the Leibniz rule $$ D(s \otimes t) = D(s) \otimes t + s \otimes D(t) \tag{2} $$ holds for any two sections $s,t$ of tensor products of a finite number of copies of the bundle $E$ and its dual $E$ (let us call them $E$-bundles), and for any contraction map $C$ we have $D \circ C = C \circ D$, we obtain a family of connections in all the $E$-bundles. The whole family of such connections is usually referred to as the connection $D$ in $E$-bundles.

The Levi-Civita connection in a Riemannina manifold $(M,g)$ is frequently denoted by $\nabla$. This is a connection in $TM$-bundles, which are better known under the name tensor bundles.

In the Riemannian setting we usually identify vectors and covectors, so the differential $\mathrm{d}f$ and the gradient $\nabla f$ are no longer distingushed, and we may write the Leibniz rule $(1)$ as $$ \nabla (f \, t) = \nabla f \otimes t + f \, \nabla t \tag{3} $$ wheres for two tensors $s,t$ we have $$ \nabla(s \otimes t) = \nabla s \otimes t + s \otimes \nabla t \tag{4} $$

Introducing another convention $f \otimes s := f s$ for a function $f \in C^{\infty}(M)$ and a tensor $s$, we may reinterpret the rule $(3)$ as a particular case of $(4)$.

Futhermore, it is also quite common to use the symbol $\nabla$ for arbitrary connections in vector bundles (instead of $D$), and taking by the convention that in this case $\nabla f := \mathrm{d}f$ for a smooth function $f$, we still have the Leibniz rule written as $(4)$, which is very convenient (the rule $(3)$ is then obtained from $(4)$ by setting $s := f$).