Let $E \rightarrow M$ be a vector bundle over a smooth manifold $M$ and let $g$ be a bundle metric. Does there exists a conection (maybe unique) $\nabla$ which is compatible with $g$. By this I mean: for two sections $s,s'$ of the bundle $E \rightarrow M$ and a vector field $X$ on $M$ we have $X(g(s,s'))=g(\nabla_{X}s,s')+g(s,\nabla_{X}s')$. I know that this is true for hermitian connection. Then the uniqueness is given by the holomorphic structure. But here I consider the smooth setting. So my question: Is there any special connection as described above? Is this connection unique? In what sense unique?

Greetings, Phillip