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Extra resources for Beyond Geometry: Classic Papers from Riemann to Einstein

Sample text

By contrast, in the natural sciences where the simple principles for such constructions are still lacking, to discover causal con­ nections one pursues phenomenon into the spatially small, just so far as the microscope permits. Questions about the metric relations of space in the im­ measurably small are thus not idle ones. If one assumes that bodies exist independently of position, then the curvature is everywhere constant, and it then follows from astronomical measurements that it cannot be different from zero; or at any rate its reciprocal must be an area in comparison with which the range of our telescopes can be neglected.

The geodesics give him a way of grasping something that lies behind mere arbitrary choice of coordinates, for the geodesics are not arbitrary. Riemann then examines whether the triangle is flat, by calculating the curvature of the plane in which it lies (the sum of its angles is > 180° if it is convex or positively curved and < 180° if concave or negatively curved). Because it is derived from geodesics (and not arbitrary lines drawn in arbitrary coordinates), the local curvature so derived does not depend on the choice of coordinates that may be used and thus is invariant.

10 There re­ main others, already completely determined by the nature of the manifold to be represented, and consequently n 2^ functions of position are required to determine its metric relations. Manifolds, like the plane and space, in which the line element can be brought into the form y ^ d x 2 thus constitute only a special case of the manifolds to be investigated here; they clearly deserve a spe­ cial name, and consequently, these manifolds, in which the square of the lines element can be expressed as the sum of the squares of complete differentials, I propose to call flat.

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