This thesis studies the construction of noncompact Einstein manifolds of cohomogeneity one on some vector bundles. Cohomogeneity one vector bundle whose isotropy representation of the principal orbit G/K has two inequivalent irreducible summands has been studied in [Böh99][Win17]. However, the method applied does not cover all cases. This thesis provides an alternative construction with a weaker assumption of G/K admits at least one invariant Einstein metric. Some new Einstein metrics of Taub-NUT type are also constructed. This thesis also provides construction of cohomogeneity one Einstein metrics for cases where G/K is a Wallach space. Specifically, two continuous families of complete smooth Einstein metrics are constructed on vector bundles over CP2, HP2 and OP2 with respective principal orbits the Wallach spaces SU(3)/T2, Sp(3)/(Sp(1)Sp(1)Sp(1)) and F4/Spin(8). The first family is a 1-parameter family of Ricci-flat metrics. All the Ricci- flat metrics constructed have asymptotically conical limits given by the metric cone over a suitable multiple of the normal Einstein metric. All the Ricci-flat metrics constructed have generic holonomy except that the complete metric with G2 holonomy discovered in [BS89][GPP90] lies in the interior of the 1-parameter family on manifold in the first case. The second family is a 2-parameter family of Poincaré–Einstein metrics. / Thesis / Doctor of Philosophy (PhD)

Dans cette thèse des actions conformes de cohomogénéité un sur l'univers d'Einstein tridimensionel sont classifiées. Notre stratégie est d'établir dans un premier temps quel peut être le groupe de transformations conformes impliqué, à conjugaison près. Nous décrivons aussi la topologie et la nature causale des orbites d'une telle action. / In this thesis, the conformal actions of cohomogeneity one on the three-dimensional Einstein universe are classified. Our strategy in this study is to determine the representation of the acting group in the group of conformal transformations of Einstein universe up to conjugacy. Also, we describe the topology and the causal character of the orbits induced by cohomogeneity one actions in Einstein universe.

In this thesis, we study the Ricci flow and Ricci soliton equations on Riemannian manifolds which admit a certain degree of symmetry. More precisely, we investigate the Ricci soliton equation on connected Riemannian manifolds, which carry a cohomogeneity one action by a compact Lie group of isometries, and the Ricci flow equation for invariant metrics on a certain class of compact and connected homogeneous spaces. In the first case, we prove that the initial value problem for a cohomogeneity one gradient Ricci soliton around a singular orbit of the group action always has a solution, under a technical assumption. However, this solution is in general not unique. This is a generalisation of the analogous result for the Einstein equation, which was proved by Eschenburg and Wang in their paper "Initial value problem for cohomogeneity one Einstein metrics". In the second case, by studying the corresponding system of nonlinear ODEs, we identify a class of singular behaviours for the homogeneous Ricci flow on these spaces. The singular behaviours that we find all correspond to type I singularities, which are modelled on rigid shrinking solitons. In the case where the isotropy representation decomposes into two invariant irreducible inequivalent summands, we also investigate the existence of ancient solutions and relate this to the existence and non existence of invariant Einstein metrics. Furthermore, in this special case, we also allow the initial metric to be pseudo- Riemannian and we investigate the existence of immortal solutions. Finally, we study the behaviour of the scalar curvature for this more general situation and show that in the Riemannian case it always has to turn positive in finite time, if it was negative initially. By contrast, in the pseudo-Riemannian case, there are certain initial conditions which preserve negativity of the scalar curvature.

Um problema central em teoria de subvariedades é estudar imersões isométricas f : Mn → Qn+kc de uma variedade Riemanniana completa em uma forma espacial sob a ação de um subgrupo conexo e fechado do grupo de isometrias Iso(M). Esse estudo teve início com o relevante trabalho de Kobayashi (KOBAYASHI, 1958), que provou que se Mn é uma hipersuperfície compacta e homogênea no espaço Euclidiano, então Mn é isométrica à esfera usual. Neste trabalho estudamos imersões isométricas em formas espaciais com codimensão igual a 2. Mais precisamente, obtemos uma classificação das imersões isométricas f : Mn → Qn+2c de uma variedade Riemanniana completa sob a ação de cohomogeneidade 1 de um subgrupo fechado G ⊂ Iso(M), de modo que as órbitas principais são hipersuperfícies umbílicas de Mn. / An important problem in submanifold theory is to study isometric immersions f : Mn → Qn+kc into a space form of a complete Riemannian manifold of dimension n acted on by a closed connected subgroup of its isometry group Iso(M). This study was initiated by Kobayashi (KOBAYASHI, 1958), who proved that if Mn is a compact and homogeneous hypersurface into Euclidean space, then Mn must be a round sphere. In this work we study isometric immersions into a space form with codimension 2. More precisely, we give a complete classification of isometric immersions f : Mn → Qn+2c of complete Riemannian manifold into a space form acted on by a closed connected subgroup G &sub: Iso(M) of cohomogeneity one, under the assumption that all principal orbits are umbilical hypersurfaces of Mn.

<p>We present an alternative proof of the existence theorem of B\"ohm using ideas from the study of gradient Ricci solitons on the multiple warped product cohomogeneity one manifolds by Dancer and Wang. We conclude that the complete Ricci-flat metric converges to a Ricci-flat cone. Also, starting from a $4n$-dimensional $\mathbb{H}P^{n}$ base space, we construct numerical Ricci-flat metrics of cohomogeneity one in ($4n+3$) dimensions whose level surfaces are $\mathbb{C}P^{2n+1}$. We show the local Ricci-flat solution is unique (up to homothety). The numerical results suggest that they all converge to Ricci-flat Ziller cone metrics even if $n=2$.</p> / Master of Science (MSc)