How can i express the scalar curvature for a one - dimensional Riemannian manifold (M, g) in terms of the metric g ?
scalar curvature on one - dimensional Riemannian Manifold
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riemannian-geometry
1 Answers
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A one-dimensional Riemannian manifold does not have any intrinsic curvature at all. It is always locally isometric to a straight, "flat" line.
Formally, the Riemann curvature tensor has but a single component $R_{1111}$, but this element is required to be 0 due to (for example) the skew-symmetry of $R_{ijkl}$.
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1Yes, the isometry is given by a unit speed geodesic. – 2015-07-22