Infinitesimal and Local Convexity of a Hypersurface in a Semi-Riemannian Manifold

Abstract

Given a Riemannian manifold (M, g) and an embedded hypersurface H in M , a result by R. L. Bishop states that infinitesimal convexity on a neighborhood of a point in H implies local convexity. Such result was extended very recently to Finsler manifolds by the author et al. [2]. We show in this note that the techniques in [2], unlike the ones in Bishop's paper, can be used to prove the same result when (M, g) is semi-Riemannian. We make some remarks for the case when only timelike, null or spacelike geodesics are involved. The notion of geometric convexity is also reviewed and some applications to geodesic connectedness of an open subset of a Lorentzian manifold are given.2000 Mathematics Subject Classification. 53C60, 53C22, 58E10.

Convex Regions of Stationary Spacetimes and Randers Spaces. Applications to Lensing and Asymptotic Flatness

“…(4) The notion of local convexity, explained in Remark 2.2, is also trivially extensible to the Lorentzian case from the Riemannian or Finslerian ones, and its equivalence with infinitesimal convexity can be also proved by transplanting the technique in [3], see [17]. Again, in the general Lorentzian case, we can define also local time-, space-or light-convexity by considering only geodesics of the corresponding type.…”

“…Again, in the general Lorentzian case, we can define also local time-, space-or light-convexity by considering only geodesics of the corresponding type. However, its equivalence with the corresponding infinitesimal notions is subtler, see [17].…”

“…(a) domains such as M a ǫ , ǫ > 0 in Kerr spacetime (see Subsection 4.3) not only are not space-convex but also are not geodesically connected [29,Corollary 3], (b) the full outer region of Schwarzschild spacetime, as well as the outer region of slow Kerr one (outside the black hole, which include the ergosphere and, so, it is not fully stationary for a = 0) is geodesically connected [30]; the proof uses different topological arguments introduced in [31], and (c) for general standard stationary spacetimes, geodesic connectedness has been studied by a combination of variational and causal methods in [16] (see also [17], for the case of domains with boundary). Even though such problems of geodesic connectedness do not have a simple physical interpretation as those of causal geodesic connectedness in this paper, they constitute and excellent arena to study critical points curves for indefinite functionals, with broad possibilities of applications.…”

“…All the conclusions of Theorem 5.1 hold if the hypothesis on the compactness forB s (p, r), is replaced by the more general hypothesis of compactness for the intrinsic ballsBD s (p, r), p ∈ D. In fact, the compactness of the setsD ∩B s (p, r) is used in Lemma 4.2 of [3] and in the proof of the Palais-Smale condition for the functionals of a family perturbing the energy functional of the Finsler manifold (see [3,Prop. 4.3]), to ensure that the supports of any sequence of absolutely continuous curves γ n : [0, 1] → D, γ n (0) = p, γ n (1) = q, p, q ∈ D, such that Our main result for lightlike geodesics is then: 10 Recall that [3, Theorem 1.3] is stated under C 2,1 loc regularity, but the result holds also for a C 2 domain (see [17]). Theorem 5.3.…”

“…Time-convexity. Randers metrics can be also used to characterize the convexity of the boundary of a (stationary) region of a standard stationary spacetime assumption can be dropped (cf [17,. Remark 3]).…”