# differential geometry – Given a covariant derivative, is there a metric tensor that has the covariant derivative as the metric connection?

In differential geometry or general relativity, we usually think of the metric tensor $$g_{mu nu}$$ first and then introduce the metric connection.

However, I wonder if we can go reverse.

That is, let $$M$$ be a smooth manifold equipped with some nontrivial affine connection $$nabla$$. Then does there always exist a metric tensor $$g_{mu nu}$$ on $$M$$ that has $$nabla$$ as its metric connection?

I tried to figure this out myself but it is more confusing than expected. Could anyone help me?