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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.

For every codimension two spacelike submanifold of a Lorentz manifold and each choice of a normal lightlike vector field, we introduce a canonical way to construct a tractor conformal bundle. We characterize when the induced connection of a such submanifold defines a tractor connection and then, in this case, when this tractor conformal bundle with the induced connection is standard for the induced metric. Finally, the normality condition for this tractor conformal bundle endowed with the induced connection is characterized in terms of a strong relationship between the intrinsic and the extrinsic geometry of the starting spacelike submanifold., Comment: 10 pages

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. Then, every metric in the conformal class corresponds to the induced metric on M seen as 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 whose Weingarten endomorphism provide a Mobius structure on the Riemannian conformal structure. Conversely, every Mobius structure on a Riemannian conformal structure arises in this way. Flat Mobius structures are characterized in terms of the extrinsic geometry of the corresponding spacelike surfaces.

The world is becoming more globalized every day and people can buy products from almost every country in the world in their local stores. Given the different food and feed safety laws from country to country, the European Union began to register in 1977 all irregularities related to traded products to ensure cross-border monitoring of information and a quick reaction when risks to public health are detected in the food chain. This information has also an enormous potential as a preventive tool, in order to warn actors involved in food safety and optimize their resources. In this paper, a set of data related to food issues was scraped and analysed with Machine Learning techniques to predict some features of future notifications, so that pre-emptive measures can be taken. The novelty of the work relies on two points: the use of categorical embeddings with Deep Learning models (Multilayer Perceptron and 1-Dimension Convolutional Neural Networks) and its application to solve the problem of predicting food issues in the European Union. The models allow several features to be predicted: product category, hazard category and finally the proper action to be taken. Results show that the system can predict these features with an accuracy ranging from 74.08% to 93.06%., Comment: 20 pages,8 figures, 15 tables