The equivalence of smooth and synthetic notions of timelike sectional curvature bounds

Timelike sectional curvature bounds play an important role in spacetime geometry, both for the understanding of classical smooth spacetimes and for the study of Lorentzian (pre-)length spaces introduced by Kunzinger and Sämann [Ann. Global Anal. Geom. 54 (2018), pp. 399-447]. In the smooth setting, a bound on the sectional curvature of timelike planes can be formulated via the Riemann curvature tensor. In the synthetic setting, bounds are formulated by comparing various geometric configurations to the corresponding ones in constant curvature spaces. The first link between these notions in the Lorentzian context was established by Harris [Indiana Univ. Math. J. 31 (1982), pp. 289–308], which was instrumental in the proof of powerful results in spacetime geometry (see Beem et al. [ <italic>Toponogov splitting theorem for Lorentzian manifolds</italic> , Springer, Berlin, 1985; J. Differential Geom. 22 (1985), pp. 29–42]; Galloway and Ling [Gen. Relativity Gravitation 50 (2018), p. 7]). For general semi-Riemannian manifolds, the equivalence between sectional curvature bounds and synthetic bounds was established by Alexander and Bishop [Comm. Anal. Geom. 16 (2008), pp. 251–282]; however in this approach the sectional curvatures of both timelike and spacelike planes have to be considered. In this article, we fill a gap in the literature by proving the full equivalence between sectional curvature bounds on timelike planes and synthetic timelike bounds on strongly causal spacetimes. As an essential tool, we establish Hessian comparison for the time separation and signed distance functions.