Riemannian, Pseudo-Riemannian, and Affinely Connected Manifolds
The subject of this chapter is the geometry of a manifold that has a metric, a pseudometric, or an affine connection defined on it. The dividing line between the preceding chapter and this may seem rather fine and arbitrary, since the last topic in that chapter was geodesics, and the first in this is geodesic coordinates. However, there is a fundamental difference between the two. The preceding chapter was mainly analytic, the only geometric notion being that of the distance between two points, whereas this one is mainly geometric, and even in the sense of Euclid, except that the concepts are somewhat extended and the formulation is analytic. The fundamental concepts, such as parallelism, length, curvature, and angle, are really geometric, and must be so regarded. The use of analytic methods does not detract from the geometric nature of those concepts any more than did the introduction of numerical coordinates into Euclidean geometry by Descartes. From that point of view, one of the main results of the preceding chapter, Whitehead’s theorem, serves the same purpose as Euclid’s postulate that through any two distinct points there can be drawn one and only one straight line, even though, in Whitehead’s theorem, the two points must not be too far apart.
KeywordsMetric pseudometric connection and topology geodesic or Riemannian coordinates geometry in the sense of Klein approximate congruence congruence of stars covariant derivative absolute derivative parallel transport orientability Lorentz orientability Laplacian d’Alembertian Riemann tensor Ricci tensor Riemann curvature scalar determination of the second derivatives of the metric tensor from the Riemann tensor conditions foran affine connection to be compatible with a metric intrinsic curvature of a manifold flatness Stäckel Robertson and Eisenhart conditions for separability of the wave equation
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