[1] Following the work of Dennis DeTurck and Jerry Kazdan in 1981, they began to play a significant role in the geometric analysis literature, although Idzhad Sabitov and S.Z.
[4] Relative to a fixed "background" coordinate chart (V, y), one can view (x1, ..., xn) as a collection of functions x ∘ y−1 on an open subset of Euclidean space.
[7] As was first discovered by Cornelius Lanczos in 1922, relative to a harmonic coordinate chart, the Ricci curvature is given by The fundamental aspect of this formula is that, for any fixed i and j, the first term on the right-hand side is an elliptic operator applied to the locally defined function gij.
[11] Harmonic coordinates were used by Robert Bartnik to understand the geometric properties of asymptotically flat Riemannian manifolds.
The key technical work is in the establishment of a Fredholm theory for the Laplace-Beltrami operator, when acting between certain Banach spaces of functions on M which decay at infinity.
Building upon his methods, Shigetoshi Bando, Atsushi Kasue, and Hiraku Nakajima showed that the decay of the curvature in terms of the distance from a point, together with polynomial growth of the volume of large geodesic balls and the simple-connectivity of their complements, implies the existence of asymptotically flat coordinates.
[19] A foundational result, due to Michael Anderson, is that given a smooth Riemannian manifold, any positive number α between 0 and 1 and any positive number Q, there is a number r which depends on α, on Q, on upper and lower bounds of the Ricci curvature, on the dimension, and on a positive lower bound for the injectivity radius, such that any geodesic ball of radius less than r is the domain of harmonic coordinates, relative to which the C1, α size of g and the uniform closeness of g to the Euclidean metric are both controlled by Q.
[22] The essential aspect of the proof is the analysis, via standard methods of elliptic partial differential equations, for the Lanczos formula for the Ricci curvature in a harmonic coordinate chart.
[24] Due to the nature of such "Riemannian convergence", it follows, for instance, that up to diffeomorphism there are only finitely many smooth manifolds of a given dimension which admit Riemannian metrics with a fixed bound on Ricci curvature and diameter, with a fixed positive lower bound on injectivity radius.