Let $(M,g)$ be a closed Riemannian manifold. Let $V$ be a smooth complex vector bundle over $M$. We can write $V$ as the range of an idempotent $E$ in a matrix algebra $M_n(C^\infty(M))$ acting on a trivial bundle $M\times\mathbb{C}^n$. Here $C^\infty(M)$ denotes the complex-valued functions on $M$.
Let $D$ be the trivial connection on $M\times\mathbb{C}^n$ defined by applying the de Rham differential to each entry separately. Then the expression $EDE$ defines a connection on $V$.
I see some textbooks have called the connection $EDE$ the "Levi-Civita connection" on $V$, "by analogy to the classical situation", and I am trying to understand how this analogy works. In particular, when $V$ is the tangent bundle $TM$ (ignoring the fact that this is a real vector bundle), the Levi-Civita connection depends on the metric $g$, while the expression $EDE$ does not seem to depend on $g$.
Question: What is the precise sense in which $EDE$ reduces to the "classical" Levi-Civita connection on $TM$?