A Riemannian metric assigns a smoothly varying positive definite inner product to the tangent space at every point of a smooth manifold. Christoffel and Lipschitz studied the problem of deciding when two Riemannian metrics differ only by a coordinate transformation (Cogliati 2024, pp. 721 and 731).
Riemannian Metric
See also
Compact Manifold, Harmonic Dimension Comparison Conjecture, Injectivity Domain, Line Element, Metric Tensor, Minkowski Metric, Riemannian Geometry, Riemannian Manifold, Yau Nodal Set ConjectureExplore with Wolfram|Alpha
References
Cogliati, A. "Some Remarks on the History of Ricci's Absolute Differential Calculus." Arch. Hist. Exact Sci. 78, 717-761, 2024. https://doi.org/10.1007/s00407-024-00336-2.Referenced on Wolfram|Alpha
Riemannian MetricCite this as:
Weisstein, Eric W. "Riemannian Metric." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RiemannianMetric.html