2015
DOI: 10.1007/s00208-015-1270-4
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Completeness of compact Lorentzian manifolds with abelian holonomy

Abstract: This note proves the geodesic completeness of any compact man-ifold endowed with a linear connection such that the closure of its holonomy group is compact.

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Cited by 32 publications
(53 citation statements)
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“…The condition (1.1) is equivalent to ∇U ♭ = g(∇U, ·) being symmetric, which in turn is equivalent to dU ♭ = 0. Now we can argue analogously as in [31,Proposition 8]:…”
Section: Riemannian Manifolds Satisfying the Constraintmentioning
confidence: 87%
See 1 more Smart Citation
“…The condition (1.1) is equivalent to ∇U ♭ = g(∇U, ·) being symmetric, which in turn is equivalent to dU ♭ = 0. Now we can argue analogously as in [31,Proposition 8]:…”
Section: Riemannian Manifolds Satisfying the Constraintmentioning
confidence: 87%
“…Setting F := U 0 = z −1 (0), this gives the local form of the metric (1.2). Moreover, if Z = 1 u 2 U is complete, [31,Proposition 8] shows that the flow of the lift of Z to the universal cover M of M defines a global diffeomorphism Ψ between M and R × F , where F is the universal cover of a leaf F of U ⊥ .…”
Section: Riemannian Manifolds Satisfying the Constraintmentioning
confidence: 99%
“…Another instance of this phenomenon is given in [21], where it is shown that compact Ricci-flat pp-waves are plane waves.…”
Section: Background and Main Resultsmentioning
confidence: 94%
“…Even though the completeness of trajectories of dynamical systems is a very classical topic (see the book [2] or the survey [33]), as far as we know, the condition of harmonicity (which defines potential theory as well as divergence free gradient vector fields) has not been studied in this setting. Indeed, the progress along these decades has focused on other aspects of the conjecture rather than in the crude geodesic equation (or the dynamical system (4)), concretely: (a) All plane waves (gravitational or not) are geodesically complete, as the equation (4) reduces to a second order linear system of differential equations [17], [ [26] gave a local characterization of pp-waves, argued that a natural extension of the conjecture follows when a locally pp-wave metric is taken on a compact M , and proved that this extension becomes equivalent to the standard one on R 4 . In this same line, the quoted article [16] also showed that plane waves are the universal coverings of certain spacetimes with non-trivial topology under some natural hypotheses for EK conjecture.…”
Section: 2mentioning
confidence: 99%