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It was long accepted as a fact that a metrical character could be desc — Bernhard Riemann

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"It was long accepted as a fact that a metrical character could be described by means of a quadratic differential form, but the fact was not clearly understood. Riemann many years ago pointed out that the metrical groundform might, with equal right, essentially, be a homogeneous function of the fourth order in the differentials, or even a function built up in some other way. and that it need not even depend rationally on the differentials. But we dare not stop even at that point."
Bernhard RiemannBernhard Riemann
Bernhard Riemann
Bernhard Riemann
Bernhard Riemann
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Georg Friedrich Bernhard Riemann was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces,

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"Definite portions of a manifoldness, distinguished by a mark or by a boundary, are called Quanta. Their comparison with regard to quantity is accomplished in the case of discrete magnitudes by counting, in the case of continuous magnitudes by measuring. Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case also we can only determine the more or less and not the how much. The researches which can in this case be instituted about them form a general division of the science of magnitude in which magnitudes are regarded not as existing independently of position and not as expressible in terms of a unit, but as regions in a manifoldness."
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"For Space, when the position of points is expressed by rectilinear co-ordinates, ds = \sqrt{ \sum (dx)^2 }; Space is therefore included in this simplest case. The next case in simplicity includes those manifoldnesses in which the line-element may be expressed as the fourth root of a quartic differential expression. ...I restrict myself... to those manifoldnesses in which the line element is expressed as the square root of a quadric differential expression. ...Manifoldnesses in which, as in the Plane and in Space, the line-element may be reduced to the form \sqrt{ \sum (dx)^2 }, are... only a particular case of the manifoldnesses to be here investigated; they require a special name, and therefore these manifoldnesses... I will call flat. In order now to review the true varieties of all the continua which may be represented in the assumed form, it is necessary to get rid of difficulties arising from the mode of representation, which is accomplished by choosing the variables in accordance with a certain principle."
Bernhard RiemannBernhard Riemann