Metric and Geodesics on a Manifold
A manifold, as defined in Chapter 23, is a thing characterized just by its local topology: It is a locally n-dimensional space in which the Hausdorff separation axiom holds. In this chapter and the next two, a manifold is made into a geometric structure by introducing further notions such as geodesics (a geodesic is the analogue of a straight line in Euclidean geometry), lengths, angles, and so on. The most fundamental notion is geodesic, which is derived, in the main geometries of interest to physics, either from a metric or from an affine connection; we start with a metric, because of its similarity with distance in familiar Euclidean geometry.
KeywordsScalar vector and tensor fields Lie brackets covariant and contravariant vectors transformation laws inner and outer multiplication contraction quotient law derivations metric tensor definite and indefinite metric Riemannian and pseudo-Riemannian manifolds raising and lowering of indices geodesics Euler variational equation natural affine or preferred parameter Christoffel three-index symbols spacelike null and timelike geodesies initial-value and two-point problems of geodesies Volterra integral equations Picard iterations Whitehead’s theorem continuation of geodesies affinely connected manifolds Riemannian and pseudo-Riemannian covering manifolds
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