What exactly does $\Delta f = \sum_{i=1}^N e_i(e_i f)$ (for all $f\in C^\infty(M)$) mean and what can we say about $N$ and $e_i$, if $M=\Bbb S^n$?

Question

When dealing with Riemannian manifolds, I often come across the following assumption:

Let $M$ a Riemannian manifold. Let $e_i$ be vector fields on $M$ such that for any smooth $f\colon M \to \Bbb R$, the Laplacian can be expressed as $\Delta f = \sum_{i=1}^N e_i(e_i f)$.

I see how in the case $M=\Bbb R^n$, I can just choose $n=N$ and $e_i=\partial_{x_i}$, but I do not really understand the intuition of this assumption in a general situation.

1. What does this assumption mean and why is it useful?
2. What does the value of $N$ tell us?
3. In a simple situation such as $M=\Bbb S^n$ (the $n$-sphere), is there an explicit choice of such $e_i$?

Answer 1

If you take a Euclidean plane in curvilinear coordinates you see that $\triangle=e_1e_1+e_2e_2+ae_1+be_2$