In this article we obtain the rigid isotopy classification of generic rational curves of degre 5 in \({\mathbb {R}}{\mathbb {P}}^{2}\). In order to study the rigid isotopy classes of nodal rational curves of degree 5 in \({\mathbb {R}}{\mathbb {P}}^{2}\), we associate to every real rational nodal quintic curve with a marked real nodal point a nodal trigonal curve in the Hirzebruch surface \(\Sigma

We study smooth rational closed embeddings of the real affine line into the real affine plane, that is algebraic rational maps from the real affine line to the real affine plane which induce smooth closed embeddings of the real euclidean line into the real euclidean plane. We consider these up to equivalence under the group of birational automorphisms of the real affine plane which are diffeomorphisms

Let X be a smooth complex projective curve, and let \(x\,\in \, X\) be a point. We compute the automorphism group of the moduli space of framed vector bundles on X of rank \(r\, \ge \, 2\) with a framing over x. It is shown that this automorphism group is generated by the following three: (1) pullbacks using automorphisms of the curve X that fix the marked point x, (2) tensorization with certain line

We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying comparable bounds. This kind of discs naturally come up in the context of the solution

We give sharp upper bounds on the injectivity radii of complete hyperbolic surfaces of finite area with some geodesic boundary components. The given bounds are over all such surfaces with any fixed topology; in particular, boundary lengths are not fixed. This extends the first author’s earlier result to the with-boundary setting. In the second part of the paper we comment on another direction for extending

We prove that any Fano manifold of coindex three admitting nef tangent bundle is homogeneous.

Given a simple closed curve \(\gamma \) on a connected, oriented, closed surface S of negative Euler characteristic, Mirzakhani showed that the set of points in the moduli space of hyperbolic structures on S having a simple closed geodesic of length L of the same topological type as \(\gamma \) equidistributes with respect to a natural probability measure as \(L \rightarrow \infty \). We prove several

This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi-isometric classification with the bi-Lipschitz classification. On the other

Tête-à-tête graphs were introduced by N. A’Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed tête-à-tête graphs provide a generalization which define mixed tête-à-tête twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed tête-à-tête twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed

Mark and Paupert devised a general method for obtaining presentations for arithmetic non-cocompact lattices, \(\Gamma \), in isometry groups of negatively curved symmetric spaces. The method involves a classical theorem of Macbeath applied to a \(\Gamma \)-invariant covering by horoballs of the negatively curved symmetric space upon which \(\Gamma \) acts. In this paper, we will discuss the application

Let G be a compact semisimple linear Lie group. We study the action of \(\text {Aut}(F_r)\) on the space \(H_*(G^r; {\mathbb {Q}})\). We compute the image of this representation and prove that it only depends on the rank of \({\mathfrak {g}}\). We show that the kernel of this representation is always the Torrelli subgroup \(\text {IA}_r\) of \(\text {Aut}(F_r)\).

We define flag structures on a real three manifold M as the choice of two complex lines on the complexified tangent space at each point of M. We suppose that the plane field defined by the complex lines is a contact plane and construct an adapted connection on an appropriate principal bundle. This includes path geometries and CR structures as special cases. We prove that the null curvature models are

Let G be a connected Lie group and \(\varGamma \subset G\) a lattice. Connection curves of the homogeneous space \(M=G/\varGamma \) are the orbits of one parameter subgroups of G. To block a pair of points \(m_1,m_2 \in M\) is to find a finite set \(B \subset M{\setminus } \{m_1, m_2 \}\) such that every connecting curve joining \(m_1\) and \(m_2\) intersects B. The homogeneous space M is blockable

We construct in projective differential geometry of the real dimension 2 higher symmetry algebra of the symplectic Dirac operator acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds

It is shown that various questions about the existence of simple closed curves in normal subgroups of surface groups are undecidable.

Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is \(\mathrm {CAT}(0)\) with a metric for which all vertex stars are convex. This strengthens and generalizes a result by Crowley. Further consequences of our work are: (1) All \(\mathrm {CAT}(0)\) cube complexes are collapsible. (2) Any triangulated

Let \(S_{g,n}\) be a surface of genus \(g > 1\) with \(n>0\) punctures equipped with a complete hyperbolic cusp metric. Then it can be uniquely realized as the boundary metric of an ideal Fuchsian polyhedron. In the present paper we give a new variational proof of this result. We also give an alternative proof of the existence and uniqueness of a hyperbolic polyhedral metric with prescribed curvature

The aim of this paper is to classify the cohomogeneity one conformal actions on the 3-dimensional Einstein universe \(\mathbb {E}{\mathrm {in}}^{1,2}\), up to orbit equivalence. In a recent paper (Hassani in C R Acad Sci Paris Ser I 355:1133–1137, 2017. https://doi.org/10.1016/j.crma.2017.10.003), we studied the unique (up to conjugacy) irreducible action of \({\mathrm {PSL}}(2,\mathbb {R})\) on \(\mathbb

We prove a partial converse to the main theorem of the author’s previous paper Proper affine actions: a sufficient criterion (submitted; available at arXiv:1612.08942). More precisely, let G be a semisimple real Lie group with a representation \(\rho \) on a finite-dimensional real vector space V, that does not satisfy the criterion from the previous paper. Assuming that \(\rho \) is irreducible and

A rigid set in a curve complex of a surface is a subcomplex such that every locally injective simplicial map from the set into the curve complex is induced by a homeomorphism of the surface. In this paper, we find finite rigid sets in the curve complexes of connected nonorientable surfaces of genus g with n holes for \(g+n \ne 4\).

Let S be a compact, orientable surface of hyperbolic type. Let \((k_+,k_-)\) be a pair of negative numbers and let \((g_+, g_-)\) be a pair of marked metrics over S of constant curvature equal to \(k_+\) and \(k_-\) respectively. Using a functional introduced by Bonsante, Mondello and Schlenker, we show that there exists a unique affine deformation \(\Gamma :=(\rho ,\tau )\) of a Fuchsian group such

We construct sequences of ‘expander manifolds’ and we use them to show that there is a complete connected 2-dimensional Riemannian manifold with discontinuous isoperimetric profile, answering a question of Nardulli and Pansu. Using expander manifolds in dimension 3 we show that for any \(\epsilon , M>0\) there is a Riemannian 3-sphere S of volume 1, such that any (not necessarily connected) surface

We introduce a new fundamental domain \(\mathscr {R}_n\) for a cusp stabilizer of a Hilbert modular group \(\Gamma \) over a real quadratic field \(K=\mathbb {Q}(\sqrt{n})\). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of \(\mathcal {H}^2\times \mathcal {H}^2\). The region \(\mathscr {R}_n\) is the product of \(\mathbb {R}^+\)

We determine the optimal horoball packings of the asymptotic or Koszul-type Coxeter simplex tilings of hyperbolic 5-space, where the symmetries of the packings are derived from Coxeter groups. The packing density \(\varTheta = \frac{5}{7 \zeta (3)} \approx 0.5942196502\ldots \) is optimal and realized in eleven cases in a commensurability class of arithmetic Coxeter tilings. For the optimal packing

Due to Janet–Cartan’s theorem, any analytic Riemannian manifolds can be locally isometrically embedded into a sufficiently high dimensional Euclidean space. However, for an individual Riemannian manifold (M, g), it is in general hard to determine the least dimensional Euclidean space into which (M, g) can be locally isometrically embedded, even in the case where (M, g) is homogeneous. In this paper

We determine the least-area unit-volume tetrahedral tile of Euclidean space, without the constraint of Gallagher et al. that the tiling uses only orientation-preserving images of the tile. The winner remains Sommerville’s type 4v.

Motivated by the study of Killing forms on compact Riemannian manifolds of negative sectional curvature, we introduce the notion of generalized vector cross products on \({\mathbb {R}}^n\) and give their classification. Using previous results about Killing tensors on negatively curved manifolds and a new characterization of \(\mathrm {SU}(3)\)-structures in dimension 6 whose associated 3-form is Killing