Can we have a Metric manifold without a connection/covariant derivative?

Question

I am following an introduction to general relativity. In this course, first the concept of connection/covariant derivative is introduced on a smooth manifold. Then thereafter, the concept of a metric manifold is introduced.

My question is: is it possible to have a metric manifold without a connection, or does a metric always induce a connection?

Intuitively, I would guess that a metric induces a connection, because if we can "measure" distances across the manifold, that "connects" (in the intuitive sense) the tangent spaces of different points on the manifold. However I cannot verify it.

Is it true or false? And Why?

ps. I'm assuming there does not exist a theorem that a particular connection also induces a metric.

Answer 1

It's true. From the metric (assuming it's differentiable!) you can compute the Christoffel symbols, which tell you the connection.

Answer 2

A differentiable metric induces a connection in a certain sense.This connection is called the Levi Civita connection. The Levi Civita connection defined by the metric $g$ is thw connection (unique) without torsion such that $g$ is parallel for the connectiion, i.e the covariant derivative of $g$ relatively to the connection is zero.

https://en.wikipedia.org/wiki/Levi-Civita_connection