Digital Object Identifier (DOI) 10.1007/s00220-017-2837-6 Commun. Math. Phys. 351, 1177–1194 (2017) Communications in Mathematical Physics Dirac Operators on Time Flat Submanifolds with Applications Oussama Hijazi 1 , Sebastián Montiel 2 , Simon Raulot 3 1 Institut Élie Cartan, Université de Lorraine, B.P. 239, 54506 Vandœuvre-Lès-Nancy Cedex, France. E-mail: Oussama.Hijazi@univ-lorraine.fr 2 Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain. E-mail: smontiel@ugr.es 3 Laboratoire de Mathématiques R. Salem UMR 6085 CNRS, Université de Rouen, Avenue de l’Université, Technopôle du Madrillet, B.P. 12, 76801 Saint-Étienne-du-Rouvray, France. E-mail: simon.raulot@univ-rouen.fr Received: 27 April 2016 / Accepted: 19 December 2016 Published online: 9 February 2017 – © Springer-Verlag Berlin Heidelberg 2017 Abstract: In this paper, we study Dirac-type operators on time flat submanifolds in spacetimes satisfying the Einstein equations with non positive cosmological constant. We apply our results to obtain global rigidity results for n-dimensional time flat submanifolds in the Minkowski spacetime as well as in the anti-de Sitter spacetime. 1. Introduction One of the fundamental results in classical general relativity is certainly the proof of the positivity of the total energy by Schoen and Yau [SY] and Witten [Wi]. This led to the more ambitious claim to associate energy to extended, but finite, spacetime domains, i.e., at the quasi-local level. Obviously, the quasi-local data could provide a more detailed characterization of the states of the gravitational field than the global ones, so they are interesting in their own right. This is one of the reasons why the geometry of spacelike 2-surfaces  in spacetime plays a crucial role in general relativity. Several attempts in this direction have been made for example by Brown and York [BY] and Kijowski et al. [K, LY1, LY2]. These quasi-local masses depend on the norm of the mean curvature vector field H and possess some desirable properties (like non-negativity under some energy conditions). Moreover, it seems natural to ask that surfaces in the Minkowski spacetime have zero mass. But as pointed out in [OST], there exist examples of surfaces in the Minkowski spacetime with positive Kijowksi–Liu–Yau quasi-local mass. Wang and Yau [WY1, WY2] introduced a new quasi-local mass for surfaces with spacelike mean curvature vector, which prevents these pathological situations. Their definition relies on the metric of the 2-surface, the norm of H and a connection one-form α H (see Definition 1) depending on the direction of H. In particular, the condition div(α H ) = 0 (1) Sebastián Montiel was partially supported by a Spanish MEC-FEDER Grant No. MTM2011-22547.