• hyperbolic
• holonomy
• half-pipe
• developing map
• cone
• anti-de Sitter
• singularity
• transition

In this work we expose some results about geometric transitions, which have been achieved by Jeffrey Danciger in his thesis "geometric transitions: from hyperbolic to AdS geometry". More precisely, we'll talk about geometric structures, and in particular we'll focus on deformations between two kinds of such structures (namely hyperbolic and anti-de Sitter) on closed manifolds, mainly in dimension 3 and possibly with a set of singular points. This is done by passing through an intermediate geometry, called half-pipe: we'll prove a converse of this phenomenon, by giving a sufficient condition (which basically involves the representations of a fundamental group in a Lie group of matrices) for recovering a path of hyperbolic and anti-de Sitter structures starting from a half-pipe one.

In Chapter 1 we'll give some general definitions about geometric structures: given a smooth n-manifold $X$ and an analytic group $G$ of diffeomorphisms of $X$, we define a $(X,G)-$structure on a $n$-manifold $M$ as an atlas with values in $X$ whose changes of coordinates locally coincide with elements of $G$, and we observe that such a structure is fully described by a developing map $D:\widetilde{M}\rightarrow X$ and a holonomy representation $\rho:\pi_1(M) \rightarrow G$ (we'll actually work on these two objects in most of the proofs). We'll be mainly interested in the cases where $X$ is either the hyperbolic space $\mathbb{H}^n$ or the anti-de Sitter space $\text{AdS}^n$, and $G$ is its isometry group: a deformation from one space to the other can be constructed by considering a continuous family of pairs $(\mathbb{X}_s,G_s)$, with $X_s \subset \mathbb{P}^n \mathbb{R}$ and $G_s<\text{PGL}(n+1,\mathbb{R})$, which is a model for either $\mathbb{H}^n$ or $\text{AdS}^n$, depending on the sign of $s$; the case $s=0$ gives the so called half-pipe geometry $(\text{HP}^n,G_{\text{HP}})$ that will be described in Chapter 2. We can then consider geometric transitions on a generic $n$-manifold $M$, i.e. a path of $(\mathbb{X}_t,G_t)-$structures with $t$ varying continuously from positive to negative values.

Since Mostow's rigidity theorem does not allow deformations of hyperbolic structures in dimension $n \ge 3$, we introduce the notion of cone singularity, so that we'll be working on geometric structures with a singular locus of codimension $2$, where a sufficiently small neighbourhood of any singular point is modeled on a cone-like space. The main goal is to prove the following regeneration theorem: if a closed $3-$manifold $N$ has a half-pipe structure $(D_{\text{HP}},\rho_{\text{HP}})$ with singular locus $\Sigma$, and $(\rho_t)$ is a suitable path of holonomy representations, then it's possible to construct a path of hyperbolic/anti-deSitter structures $(D_t,\rho_t)$ limiting to $(D_{\text{HP}},\rho_{\text{HP}})$. We'll first prove this result separately on a tubular neighbourhood of the singular locus and on its complementary in $N$, and then we'll construct a geometric transition on the whole $N$.

Chapter 3 will give an alternative model for the spaces $\mathbb{H}^3$, $\text{AdS}^3$ and $\text{HP}^3$ by defining $\mathbb{P}^3\mathbb{R}$ as a projectivized space of hermitian $2 \times 2$ matrices with coefficients in a $2-$dimensional $\mathbb{R}-$algebra, so that it will be possible to restate the previous theorem. Furthermore, we'll see that the existence of a path of holonomies is implied by a smoothness condition about the representation variety $R(\pi_1(M \setminus \Sigma),\text{PSL}(2,\mathbb{R}))$. We'll finally apply this regeneration theorem to a half-pipe structure that will be constructed in Chapter 4.