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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.

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    $\begingroup$ The "birth background" is Gauss's 1825-1827 works on differential geometry of surfaces, where he introduced the first fundamental form and proved that all "measurements on a surface" (intrinsic geometry) are determined by it. This form was a 2D version of the metric tensor, which Riemann later divorced from embedded surfaces and generalized to higher dimensions. Gauss already extracted geodesics and sectional curvatures from it. Dombrowski translated Gauss's works into English, you can read them along with his extensive commentary here. $\endgroup$
    – Conifold
    Commented Oct 3, 2023 at 13:18

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The metric tensor was introduced by Riemann in his lecture: 1854 – Über die Hypothesen, welche der Geometrie zugrunde liegen. There are several English translations, and a comprehensive commentary in the book by M. Spivak, A comprehensive introduction to differential geometry, vol. I.

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  • $\begingroup$ The lecture was 1854: In 1868 Riemann was dead for two years. $\endgroup$
    – Moishe Kohan
    Commented Oct 5, 2023 at 15:47

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