Some regularity theorems in riemannian geometry

DeTurck, Dennis; Kazdan, Jerry L.

doi:10.24033/asens.1405

Cited by 284 publications

(206 citation statements)

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“…Harmonic coordinates are optimal [12] in the sense that if the metric is of class C k,γ in certain coordinates then it has at least the same regularity in harmonic coordinates.…”

Section: Earlier Workmentioning

confidence: 99%

Injectivity Radius of Lorentzian Manifolds

Chen

LeFloch

2008

Commun. Math. Phys.

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Motivated by the application to spacetimes of general relativity we investigate the geometry and regularity of Lorentzian manifolds under certain curvature and volume bounds. We establish several injectivity radius estimates at a point or on the past null cone of a point. Our estimates are entirely local and geometric, and are formulated via a reference Riemannian metric that we canonically associate with a given observer (p, T) -where p is a point of the manifold and T is a future-oriented time-like unit vector prescribed at p. The proofs are based on a generalization of arguments from Riemannian geometry. We first establish estimates on the reference Riemannian metric, and then express them in term of the Lorentzian metric. In the context of general relativity, our estimates should be useful to investigate the regularity of spacetimes satisfying Einstein field equations.

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“…Harmonic coordinates are optimal [12] in the sense that if the metric is of class C k,γ in certain coordinates then it has at least the same regularity in harmonic coordinates.…”

Section: Earlier Workmentioning

confidence: 99%

Injectivity Radius of Lorentzian Manifolds

Chen

LeFloch

2008

Commun. Math. Phys.

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“…Since this is all that we need, all of our arguments still work. Also, all of the arguments used in [ 10 and 9] to construct almost linear and harmonic coordinates carry through to prove the following (also see [5]):…”

Section: Ectxmmentioning

confidence: 99%

Removing Point Singularities of Riemannian Manifolds

Smith¹,

Yang²

1992

Transactions of the American Mathematical Society

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Abstract. We study the behavior of geodesies passing through a point singularity of a Riemannian manifold. In particular, we show that if the curvature does not blow up too rapidly near the singularity, then the singularity is at worst an orbifold singularity. The idea is to construct the exponential map centered at a singularity. Since there is no tangent space at the singularity, a surrogate is needed. We show that the vector space of radially parallel vector fields is well defined and that there is a correspondence between unit radially parallel vector fields and geodesies emanating from the singular point.

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“…odV y = 27rx(^ (4) Some regularity results relating to this work, as well as special results for Einstein manifolds will appear in [2].…”

Section: A Necessary and Sufficient Condition That R Is Locally The Rmentioning

confidence: 99%