I was usually interested in metric tensor, So I have hard searched it, But most of what I was looking for was about the 'theory of relativity'. Even so, I can find information about mathematical content, Poincare disc model of Hyperbolic geometry.
In the 'Poincare disc model' I can think of the metric tensor's meaning which is to apply Pythagoras' theorem to a curved situation. However, I am not sure about the mathematical meaning of the metric tensor yet. I know it was Riemann who first discovered this theory in the era before Einstein's application of the metric sensor to his theory of relativity. I want to know about Riemann's thinking for making metric tensor.
In other words, I want to know the overall birth background of the metric tensor. i.e. I want to know whether metric tensor can be induced purely mathematically or not.