metric tensors in simple words

Question

I have noticed manifold spaces is defined with a metric tensor. What is the role of a metric tensor in Manifold spaces, in simple words? Is it to define the notion of distance between two points on the manifold?

(I have no background on differential geometry or Manifold optimization, but need to understand an instance of it.)

Answer 1

Yes, it is exactly that: to measure distances. See Section E on Page 81 in Mathematical Methods in Classical Mechanics.