Spacelike immersions in certain Lorentzian manifolds with lightlike foliations

The generalized Schwarzschild spacetimes are introduced as warped manifolds where the base is an open subset of $\mathbb{R}^2$ equipped with a Lorentzian metric and the fiber is a Riemannian manifold. This family includes physically relevant spacetimes closely related to models of black holes. The generalized Schwarzschild spacetimes are endowed with involutive distributions which provide foliations by lightlike hypersurfaces. In this paper, we study spacelike submanifolds immersed in the generalized Schwarzschild spacetimes, mainly, under the assumption that such submanifolds lie in a leaf of the above foliations. In this scenario, we provide an explicit formula for the mean curvature vector field and establish relationships between the extrinsic and intrinsic geometry of the submanifolds. We have derived several characterizations of the slices, and we delve into the specific case where the warping function is the radial coordinate in detail. This subfamily includes the Schwarzschild and Reissner-Nordstr\"om spacetimes

M\"{o}bius structures and Weingarten endomorphisms of spacelike submanifolds

Starting from a Riemannian conformal structure on a manifold $M$, we provide a method to construct a family of Lorentzian manifolds. The construction relies on the choice of a metric in the conformal class and a smooth $1$-parameter family of self-adjoint tensor fields and it has been inspired by the Fefferman and Graham ambient metric for conformal structures. Every metric in the conformal class corresponds to the induced metric on a codimension two spacelike submanifold into these Lorentzian manifolds. Under suitable choices of the $1$-parameter family of tensor fields, there exists a lightlike normal vector field along such spacelike submanifolds which Weingarten endomorphisms provide a M\"{o}bius structure on the Riemannian conformal structure. Conversely, every M\"{o}bius structure on a Riemannian conformal structure arises in this way. The Ricci-flatness condition along the scale bundle as a lightlike hypersurface into these Lorentzian manifolds is studied by means of the initial velocity of the $1$-parameter family of self-adjoint tensor fields. Finally, flat M\"{o}bius structures are characterized in terms of the extrinsic geometry of the corresponding spacelike surfaces

Rigidity results for complete spacelike submanifolds in plane fronted waves

New rigidity results for complete non-compact spacelike submanifolds of arbitrary codimension in plane fronted waves are obtained. Under appropriate assumptions, we prove that a complete spacelike submanifold in these spacetimes is contained in a characteristic lightlike hypersurface. Moreover, for a complete codimension two extremal submanifold in a plane fronted wave we show sufficient conditions to guarantee that it is a (totally geodesic) wavefront.Spanish Government PID2020-118452GBI00 PID2020-116126GB-I00ERDF project P20-01391"Maria de Maeztu" Excellence Unit IMAG - MCIN-AEI CEX2020-001105-

Chained Orchestrator Algorithm for RAN-Slicing Resource Management: A Contribution to Ultra-Reliable 6G Communications

The exponentially growing trend of Internet-connected devices and the development of new applications have led to an increase in demands and data rates flowing over cellular networks. If this continues to have the same tendency, the classification of 5G services must evolve to encompass emerging communications. The advent of the 6G Communications concept takes this into account and raises a new classification of services. In addition, an increase in network specifications was established. To meet these new requirements, enabling technologies are used to augment and manage Radio Access Network (RAN) resources. One of the most important mechanisms is the logical segmentation of the RAN, i.e. RAN-Slicing. In this study, we explored the problem of resource allocation in a RAN-Slicing environment for 6G ecosystems in depth, with a focus on network reliability. We also propose a chained orchestrator algorithm for dynamic resource management that includes estimation techniques, inter-slice resource sharing and intra-slice resource assignment. These mechanisms are applied to new types of services in the future generation of cellular networks to improve the network latency, capacity and reliability. The numerical results show a reduction in blocked connections of 38.46% for eURLLC type services, 21.87% for feMBB services, 12.5% for umMTC, 11.86% for ELDP and 11.76% for LDHMC.Spanish National Program of Research, Development, Innovation, under Grant RTI2018-102002-A-I00Junta de Extremadura under Project IB18003 and Grant GR2109