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Properly speaking, geometry is the study of manifolds that are equipped with some additional structure that permits measurements. For example, nowhere in the definition of a piecewise smooth curve is there anything that would enable us to measure the length of the curve. Likewise, on a compact, oriented n-manifold, we can integrate n-forms, but which of these integrals should be interpreted as the volume of the manifold? And given intersecting curves, how could we measure the angle they make at an intersection point? The additional structure that is needed is a metric tensor, Riemannian metrics and, to a lesser extent, pseudo-Riemannian metrics, being the main examples.
KeywordsRiemannian Manifold Symmetric Space RIEMANNIAN Geometry Parallel Transport Complete Riemannian Manifold
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