Paul E. Ehrlich

Line Integration of Ricci Curvature and Conjugate Points in Lorentzian and Riemannian Manifolds.

Generalizing results of Cohn-Vossen and Gromoll, Meyer for Riemannian manifolds and Hawking and Penrose for Lorentzian manifolds, we use Morse index theory techniques to show that if the integral of the Ricci curvature of the tangent vector field of a complete geodesic in a Riemannian manifold or of a complete nonspacelike geodesic in a Lorentzian manifold is positive, then the geodesic contains a pair of conjugate points. Applications are given to geodesic incompleteness theorems for Lorentzian manifolds, the end structure of complete noncompact Riemannian manifolds, and the geodesic flow of compact Riemannian manifolds.

Cut points, conjugate points and Lorentzian comparison theorems

1. Introduction . The purpose of this paper is to study the global geometry of a space–time ( M, g ), which is related to the Lorentzian distance function induced on the manifold M by the Lorentzian structure. We will use the signature convention (−, +, …, +) for g and assume that ( M, g ) is time orientated. The first part of this paper deals with cut points and maximal geodesies, both of which were defined in (1) using the Lorentzian distance function in analogy to the standard concepts in Rie-mannian geometry. In (1), sections 2 and 3, some elementary properties of maximal geodesies were established. In particular, the principle that, for strongly causal space-times, a limit curve of a sequence of future-directed nonspacelike ‘almost maximal’ curves is a maximal geodesic was used to prove nonspacelike incompleteness ((1), theorem 6·3). Also null cut points were used to obtain results on null incompleteness ((1), section 5). In (2) we studied deeper properties of maximal geodesies and cut points using the technical tools developed in (1), sections 2 and 3. The first part of the present paper continues these investigations.

Stability of geodesic incompleteness for Robertson-Walker space-times

Let (M, g) be a Lorentzian warped product space-timeM=(a, b)×H, g = −dt2 ⊕fh, where −∞⩽a<b⩽+∞, (H, h) is a Riemannian manifold andf: (a, b)→(0, ∞) is a smooth function. We show that ifa>−∞ and (H, h) is homogeneous, then the past incompleteness of every timelike geodesic of (M,g) is stable under smallC0 perturbations in the space Lor(M) of Lorentzian metrics forM. Also we show that if (H,h) is isotropic and (M,g) contains a past-inextendible, past-incomplete null geodesic, then the past incompleteness of all null geodesics is stable under smallC1 perturbations in Lor(M). Given either the isotropy or homogeneity of the Riemannian factor, the background space-time (M,g) is globally hyperbolic. The results of this paper, in particular, answer a question raised by D. Lerner for big bang Robertson-Walker cosmological models affirmatively.

The space-time cut locus

Let (M, g) be a space-time with Lorentzian distance functiond. If (M, g) is distinguishing andd is continuous, then (M, g) is shown to be causally continuous. Furthermore, a strongly causal space-time (M, g) is globally hyperbolic iff the Lorentzian distance is always finite valued for all metricsg′ conformal tog. Lorentzian distance may be used to define cut points for space-times and the analogs of a number of results holding for Riemannian cut loci may be established for space-time cut loci. For instance in a globally hyperbolic space-time, any timelike (or respectively, null) cut pointq of p along the geodesicc must be either the first conjugate point ofp or else there must be at least two maximal timelike (respectively, null) geodesics fromp toq. Ifq is a closest cut point ofp in a globally hyperbolic space-time, then eitherq is conjugate top or elseq is a null cut point. In globally hyperbolic space-times, no point has a farthest nonspacelike cut point.

Incompleteness of timelike submanifolds with nonvanishing second fundamental form

LetM be a properly immersed timelike hypersurface of Minkowski space and assume thatM has a strictly positive second fundamental form. If each point ofM is of diagonal type and dimM ⩾ 3, then the Ricci curvature ofM is strictly positive on all (nonzero) nonspacelike vectors. ThusM satisfies both the generic and strong energy conditions and a singularity theorem forM may be established.

Conformal deformations, Ricci curvature and energy conditions on globally hyperbolic spacetimes

We consider globally hyperbolic spacetimes ( M, g ) of dimension ≥ 3 satisfying the curvature condition Ric ( g ) ( v, v ) ≥ 0 for all non-spacelike tangent vectors v in TM . This curvature condition arises naturally as an energy condition in cosmology. Suppose ( M, g ) admits a smooth globally hyperbolic time function h : M → such that for some t 0 , the Cauchy surface h −1 ( t 0 ) satisfies the strict curvature condition Ric ( g ) ( v, v ) > 0 for all non-spacelike v attached to h −1 ( t 0 ). Then M admits a metric g ′ conformal to g satisfying the strict curvature condition Ric ( g ′) ( v, v ) > 0 for all non-spacelike v in TM . If the curvature and strict curvature conditions are restricted to null vectors, the analogous result may be obtained. Similar results may also be obtained for the scalar curvature in dimension ≥ 2 and for non-positive Ricci curvature.

Gradient-like and integrable vector fields on ℝ2

Recently, the authors have obtained criteria for the integral curves of a nonsingular smooth vector field X on a smooth manifold M to be timelike, null or spacelike geodesics for some Lorentzian metric g for M. In this paper, we show that for smoothly contractible subsets S of ℝ2 null geodesibility of a vector field X is equivalent to X being preHamiltonian on S and timelike, spacelike or Riemannian pregeodesibility of X are all equivalent to X being gradient-like. It turns out that null geodesibility is quite rare as we prove that even among real analytic vector fields on S there are many open sets of vector fields which fail to be preHamiltonian.

Constructing maximal geodesics in strongly causal space-times

Let ( M, g ) be an arbitrary space-time of dimension ≥ 2 and let d = d ( g ): M × M → ℝ ∪ {∞} (where d ( p, q ) = 0 for q ∉ J + ( p )) denote the Lorentzian distance function of ( M, g ). Also let C ( M, g ) denote the space of Lorentzian metrics for M globally con-formal to g . Here g 1 is said to be globally conformal to g if there exists a smooth function Ω: M → (0, ∞) such that g 1 = Ω g .