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Spacelike Submanifolds, Their Umbilical Properties and Applications to Gravitational Physics

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Spacelike Submanifolds, Their Umbilical Properties and Applications to Gravitational Physics Spacelike submanifolds, their umbilical properties and applications to gravitational physics Spacelike submanifolds, their umbilical properties and applications to gravitational physics Nastassja Cipriani Department of Mathematics KU Leuven Department of Theoretical Physics and History of Science University of the Basque Country October 2017 (c)2017 NASTASSJA CIPRIANI Supervisors: Prof. dr. José María Martín Senovilla Prof. dr. Joeri Van der Veken Thesis committee: Prof. dr. Stefaan Poedts, KU Leuven (president) Prof. dr. José María Martín Senovilla, University of the Basque Country (supervisor) Prof. dr. Joeri Van der Veken, KU Leuven (supervisor) Prof. dr. Keti Tenenblat, University of Brasília Prof. dr. Antoine Van Proeyen, KU Leuven Prof. dr. Raül Vera, University of the Basque Country Prof. dr. Luc Vrancken, KU Leuven and Université de Valenciennes et du Hainaut-Cambrésis Dedico questa tesi alle mie nonne Angela Boccabella e Lina Cherubini per essere state, a loro insaputa, un esempio. “You can’t easily fit women into a structure that is already coded as male, you have got to change the structure.” Mary Beard, University of Cambridge Lecture delivered at the British Museum on 3 March 2017 Acknowledgments I would like to thank the Department of Mathematics of KU Leuven, my supervisor pro- fessor Joeri Van der Veken and all current members of the geometry section, in particular Marco Zambon and my first officemate Paco. I would also like to thank Alberto, who joined me since the very first week and is now one of my best friends; and then Luca, Margherita, Akos, Gabor and Reka, Tomer, Madison, Elisa and Marta. Quiero agradecer al Departamento de Física Teórica e Historia de la Ciencia de la Uni- versidad del País Vasco, a mi director de tesis profesor José María Martín Senovilla, a Raül Vera y a todas aquellas personas que en la UPV/EHU me han hecho sentir como en casa. En particular a Borja, Pablo y João, mis frescas Olatz, María, Naiara y Laura, Charlotte, Kepa, Raúl: sois para mí como una familia. Y también a Iagoba, Fran, Asier, Iker, Aitziber, Julen, Santi, Nuria, Mikel, Nora, Elena, Cristina. Ringrazio i miei genitori per non avermi mai imposto limiti; le compagne del Piccolo di Pietralata ed Elisa. Ringrazio infine Vito, per essere stato presente sempre ed avermi dato così tanto. i Abstract We give a characterization theorem for umbilical spacelike submanifolds of arbitrary di- mension and co-dimension immersed in a semi-Riemannian manifold. Letting the co- dimension arbitrary implies that the submanifold may be umbilical with respect to some subset of normal directions. This leads to the definition of umbilical space and to the study of its dimension. The trace-free part of the second fundamental form, called total shear tensor in this thesis, plays a central role in the characterization theorems. It allows us to define shear objects (shear operators, shear tensors and shear scalars) that deter- mine the umbilical properties of the spacelike submanifold with respect to a given normal vector field. Given a group of conformal motions G acting on a semi-Riemannian manifold and an orbit S, we apply the characterization results in order to find necessary and sufficient conditions for S to have a non-empty umbilical space. We prove that if the isotropy subgroup of G is trivial, then the umbilical condition depends on the scalar products of a set of generating conformal Killing vector fields. If the isotropy subgroup of G is non-trivial, we argue that, under specific assumptions, it is possible to prove that the umbilical condition is automatically satisfied so that the umbilical space is non-trivial. The assumptions would depend on the co-dimension of S, the dimension of the isotropy subgroup and the ranks of specific matrices defined in terms of the structure constants of G. In the last part of the thesis we consider Lorentzian warped products M = M ×f Y and we analyse a particular class of spacelike submanifolds S. We find a sufficient con- dition that allows us to prove, on one hand, the existence of focal points along timelike or null geodesics normal to S and, on the other hand, the null geodesic incompleteness of M under additional reasonable conditions. By assuming that we can split the immersion as S! Σ !M, where Σ is either M ×fqg or fqg×Y, we find that the Galloway-Senovilla condition [29] can be written in terms of the warping function f and the Riemann ten- sor of either only M or Y. This means that, for instance, in order to prove singularity theorems one can restrict the study to just one of the two manifolds defining the warped product rather than considering the warped product manifold itself. We translate the con- dition found to some specific situations, such as positive and constant sectional curvature, Einstein and Ricci-flat spaces and to a few subcases in terms of the co-dimension of S. The same has been done in direct products (f = 1). iii Beknopte samenvatting We geven een karakterisatiestelling voor ombilicale ruimtelijke deelvariëteiten met wil- lekeurige dimensie en codimensie van een semi-Riemannse variëteit. Omdat de codi- mensie willekeurig is, kan de deelvariëteit ombilicaal zijn ten opzichte van een deel van de normale richtingen. Dit leidt tot de definitie van ombilicale ruimte en tot de studie van de dimensie ervan. Het spoorvrije deel van het tweede fundamentaalvorm, dat we in dit proefschrift de totale sheartensor noemen, speelt een centrale rol in de karakter- isatiestellingen. Het stelt ons in staat om objecten te definiëren die gerelateerd zijn aan de totale sheartensor (shearoperatoren, sheartensoren en sheargetallen) en die de ombili- cale eigenschappen van de ruimtelijke deelvariëteit bepalen ten opzichte van een normaal vectorveld. Gegeven een groep G van conforme transformaties van een semi-Riemannse variëteit en een orbiet S, passen we de karakterisatieresultaten toe om nodige en voldoende voor- waarden te vinden opdat S een niet-triviale ombilicale ruimte zou hebben. We bewi- jzen dat als de isotropiedeelgroep van G triviaal is, de voorwaarde om ombilicaal te zijn afhangt van de onderlinge scalaire producten van een verzameling van voortbrengende conforme Killingvectorvelden. Als de isotropiedeelgroep van G niet triviaal is, dan dan kunnen we, onder bepaalde aannames over de codimensie van S, de dimensie van de isotropiedeelgroep en de rangen van bepaalde matrices gedefinieerd in termen van de structuurconstanten van G, bewijzen dat de voorwaarde om ombilicaal te zijn automa- tisch voldaan is, i.e., dat de ombilicale ruimte niet triviaal is. In het laatste deel van het proefschrift beschouwen we Lorentziaanse gekruiste pro- ducten M = M ×f Y en bestuderen we een bepaalde klasse van ruimtelijke deel- variëteiten S. We vinden een voldoende voorwaarde voor, enerzijds, het bestaan van brandpunten langs tijd- of lichtgeodeten die loodrecht op S vertrekken en, anderzijds, de lichtgeodetische onvolledigheid van M onder enkele bijkomende redelijke voorwaarden. Door aan te nemen dat we de immersie kunnen opsplitsen in een samenstelling van im- mersies S! Σ !M, waarbij Σ ofwel M × fqg ofwel fqg × Y is, vinden we dat de voorwaarde van Galloway en Senovilla [29] kan worden geschreven in termen van de functie f en de krommingstensor van enkel M of Y. Dit betekent bijvoorbeeld dat we de studie kunnen beperken tot slechts één van de factoren van het gekruist product, in plaats van het gekruist product als geheel te beschouwen. We vertalen de gevonden voorwaarde naar enkele specifieke situaties, zoals ruimten met positieve en constante sec- tionele kromming, Einsteinruimten en Ricci-platte ruimten en naar enkele deelgevallen afhankelijk van de codimensie van S. De resultaten impliceren analoge resultaten voor directe producten (f = 1). v Resumen Hasta aproximadamente el siglo dieciocho los científicos eran expertos tanto en los as- pectos matemáticos como en los aspectos físicos de su investigación y de sus descubrim- ientos. Cabe sostener que muy probablemente estos no distinguían mucho entre la física teórica y la física experimental, por ejemplo, puesto que a menudo eran al mismo tiempo matemáticos, astrónomos, geómetras, ingenieros, físicos e incluso, a veces, filósofos. A partir del siglo diecinueve las fronteras que habrán de delimitar estos campos comien- zan a ser más definidas y desde el siglo veinte matemáticos y físicos, por lo general, han trabajado en dos áreas de investigación separadas en la ciencia. Hoy en día, debido a la rápida especialización de los campos científicos y al nacimiento de nuevas disciplinas, la distancia entre las matemáticas y la física se ha hecho aún más grande. En particular, el uso de diferentes lenguajes ha empezado a impedir que los cien- tíficos se comuniquen entre ellos y compartan sus resultados. Aun así, la historia de la ciencia nos ha mostrado cómo el intercambio de conocimiento entre estos dos campos, las matemáticas y la física, ha sido clave en el pasado para superar obstáculos, para producir cambios de paradigma, para abrir nuevas perspectivas y para impulsar una revolución científica. Dos ejemplos destacables de este proceso en estas dos ramas científicas han sido los desarrollos producidos por la teoría de la mecánica cuántica y por la teoría de la relativi- dad general. Los avances requeridos por la mecánica cuántica en el análisis funcional y la no conmutatividad de los operadores que representan los observables cuánticos, han con- ducido a un campo completamente nuevo, y ahora independiente dentro de las matemáti- cas llamado geometría no conmutativa. Por lo que concierne a la relatividad general, la equivalencia de la curvatura con el campo gravitacional, que representa uno de los fun- damentos de la teoría, es una manifestación evidente de la conexión profunda entre la geometría diferencial y la física gravitacional. En ambos casos, las matemáticas proveen a los físicos de bases sólidas y marcos para construir una nueva teoría física, mientras que la física motiva e indica a los matemáticos el camino hacia áreas inexploradas y problemas irresolutos.
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