Homogeneous and curvature homogeneous Lorentzian critical metrics

We determine all three-dimensional homogeneous and 1 -curvature homogeneous Lorentzian metrics which are critical for a quadratic curvature functional. As a result, we show that any quadratic curvature functional admits different non-Einstein homogeneous critical metrics and that there exist homogeneous metrics which are critical for all quadratic curvature functionals without being EinsteinSupported by projects PID2019-105138GB-C21(AEI/FEDER, Spain) and ED431C 2019/10, ED431F 2020/04 (Xunta de Galicia, Spain)S

Classification of the relative positions between a small ellipsoid and an elliptic paraboloid

©2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/. This version of the article Brozos-Vázquez, M., Pereira-Sáez, M. J., Souto-Salorio, M. J., & Tarrío-Tobar, A. D. (2019). “Classification of the relative positions between a small ellipsoid and an elliptic paraboloid” has been accepted for publication in Computer Aided Geometric Design, 72, 34–48. The Version of Record is available online at https://doi.org/10.1016/j.cagd.2019.05.002.[Abstract]: We classify all the relative positions between an ellipsoid and an elliptic paraboloid when the ellipsoid is small in comparison with the paraboloid (small meaning that the two surfaces cannot be tangent at two points simultaneously when one is moved with respect to the other). This provides an easy way to detect contact between the two surfaces by a direct analysis of the coefficients of a fourth degree polynomial.The authors wish to thank the referees for extremely valuable comments and suggestions, which were essential to improve the final version of the paper. Supported by Projects ED431F 2017/03, TIN2017-85160-C2-1-R, MTM2016-75897-P and MTM2016-78647-P (AEI/FEDER, UE).Xunta de Galicia; ED431F 2017/0