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The idea of a Riemannian metric having curvature, while intuitively appealing and natural, is also often the stumbling block for further progress into the realm of geometry. The most elementary way of defining curvature is to set it up as an integrability condition. This indicates that when it vanishes it should be possible to solve certain differential equations, e.g., that the metric is Euclidean. This was in fact one of Riemann’s key insights.
KeywordsJacobi Field Warped Product Metric Einstein Constant Smooth Distance Function Curvature Tensor
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