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This paper discusses the connection between projective relatedness and conformal flatness for 4-dimensional manifolds admitting a metric of signature (+,+,+,+) or (+,+,+,−). It is shown that if one of the manifolds is conformally flat and not of the most general holonomy type for that signature then, in general, the connections of the manifolds involved are the same and the second manifold is also conformally flat. Counterexamples are provided which place limitations on the potential strengthening of the results.

To a hyperbolic manifold one can associate a canonical projective structure and ask whether it can be deformed or not. In a cusped manifold, one can ask about the existence of deformations that are trivial on the boundary. We prove that if the canonical projective structure of a cusped manifold is infinitesimally projectively rigid relative to the boundary, then infinitely many Dehn fillings are projectively rigid. We analyze in more detail the figure eight knot and the Withehead link exteriors, for which we can give explicit infinite families of slopes with projectively rigid Dehn fillings., Accepted for publication at GT

Y. Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many sub-manifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial results in the direction of a potential converse to Benoist's theorem. We show that a cusped hyperbolic three-manifold may, under certain assumptions, be deformed to convex projective structures with totally geodesic torus boundary. Such structures may be convexly glued together whenever the geometry at the boundary matches up. In particular, we prove that many doubles of cusped hyperbolic three-manifolds admit convex projective structures., 48 pages, 8 figures, 2 tables

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The purpose of this paper is to understand the structures of ends of real projective $n$-dimensional orbifolds. In particular, these have the radial or totally geodesic ends. Hyperbolic manifolds with cusps and hyper-ideal ends are examples. For this, we will study the natural conditions on eigenvalues of holonomy representations of ends when these ends are manageably understandable. We will show that only the radial or totally geodesic ends of lens type or horospherical ends exist for strongly irreducible properly convex real projective orbifolds under the suitable conditions. The purpose of this article is to announce these results., Comment: 53 pages, 4 figures. Some errors of the previous version are corrected, and references are updated. arXiv admin note: text overlap with arXiv:1304.1605

Veronese webs appear as the natural way of passing to the quotient of curves in the projective space. In thi paper, we give the link between classical multidimensionnal webs and veronse webs by mean of interpolation., Comment: 13 pages

We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations., 31pages, TeX

We determine the space of commuting symmetries of the Laplace operator on pseudo-Riemannian manifolds of constant curvature, and derive its algebra structure. Our construction is based on the Riemannian tractor calculus, allowing to construct a prolongation of the differential system for symmetric Killing tensors. We also discuss some aspects of its relation to projective differential geometry., 21 pages

In two papers titled "On the so-called non-Euclidean geometry", I and II, Felix Klein proposed a construction of the spaces of constant curvature -1, 0 and and 1 (that is, hyperbolic, Euclidean and spherical geometry) within the realm of projective geometry. Klein's work was inspired by ideas of Cayley who derived the distance between two points and the angle between two planes in terms of an arbitrary fixed conic in projective space. We comment on these two papers of Klein and we make relations with other works.

International audience; Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise interpretations of this problem, with various additional conditions satisfied. The most interesting solutions are probably those inspired by an integral formula that was first introduced in this theory by Herbert Busemann. Besides that, Busemann and his school made a thorough investigation of metrics defined on subsets of projective space for which the projective lines are geodesics and they obtained several results, characterizing several classes of such metrics. We review some of the developments and important results related to Hilbert's problem, especially those that arose from Busemann's work, mentioning recent results and making connections with several branches of mathematics, including Riemannian geometry, the foundations of mathematics, the calculus of variations, metric geometry and Finsler geometry.

The final version of this paper will appear in the Handbook of Hilbert geometry, published by the European Mathematical Society (A. Papadopoulos and M. Troyanov, ed.); Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise interpretations of this problem, with various additional conditions satisfied. The most interesting solutions are probably those inspired from an integral formula that was first introduced in this theory by Herbert Busemann. Besides that, Busemann and his school made a thorough investigation of metrics defined on subsets of projective space for which the projective lines are geodesics and they obtained several results, characterizing several classes of such metrics. We review some of the developments and important results related to Hilbert's problem, especially those that arose from Busemann's work, mentioning recent results and connections with several branches of mathematics, including Riemannian geometry, the foundations of mathematics, the calculus of variations, metric geometry and Finsler geometry.

This paper demonstrates the power of the calculus developed in the two previous parts of the series for all real forms of the almost Hermitian symmetric structures on smooth manifolds, including e.g. conformal Riemannian and almost quaternionic geometries. Exploiting some finite dimensional representation theory of simple Lie algebras, we give explicit formulae for distinguished invariant curved analogues of the standard operators in terms of the linear connections belonging to the structures in question, so in particular we prove their existence. Moreover, we prove that these formulae for k-th order standard operators, k=1,2,..., are universal for all geometries in question., AMS-TeX, 33 pages

While in Euclidean, equiaffine or centroaffine differential geometry there exists a unique, distinguished normalization of a regular hypersurface immersion x: Mn → An+1, in the geometry of the general affine transformation group, there only exists a distinguished class of such normalizations, the class of relative normalizations. Thus, the appropriate invariants for speaking about affine hypersurfaces are invariants of the induced classes, e.g. the conformai class of induced metrics and the projective class of induced conormal connections. The aim of this paper is to study such invariants. As an application, we reformulate the fundamental theorem of affine differential geometry.

We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In particular we are able to construct an invariant and efficient calculus for conformal almost Hermitian geometries, and also for almost complex structures that are equipped with a projective structure. In the latter case we find a projectively invariant tensor the vanishing of which is necessary and sufficient for the existence of an almost complex connection compatible with the path structure. In both the conformal and projective setting we give torsion characterisations of the canonical connections and introduce certain interesting higher order invariants., Comment: 37 pages

A strictly convex curve is a $C^{\infty}$ -regular simple closed curve whose Euclidean curvature function is positive. Fix a strictly convex curve $\Gamma$ , and take two distinct tangent lines $l_{1}$ and $l_{2}$ of $\Gamma$ . A few years ago, Brendan Foreman proved an interesting four-vertex theorem on semiosculating conics of $\Gamma$ , which are tangent to $l_{1}$ and $l_{2}$ , as a corollary of Ghys’s theorem on diffeomorphisms of $S^{1}$ . In this paper, we prove a refinement of Foreman’s result. We then prove a projectively dual version of our refinement, which is a claim about semiosculating conics passing through two fixed points on $\Gamma$ . We also show that the dual version of Foreman’s four-vertex theorem is almost equivalent to the Ghys’s theorem.

We study discrete conjugate nets whose Laplace sequence is of period four. Corresponding points of opposite nets in this cyclic sequence have equal osculating planes in different net directions, that is, they correspond in an asymptotic transformation. We show that this implies that the connecting lines of corresponding points form a discrete W-congruence. We derive some properties of discrete Laplace cycles of period four and describe two explicit methods for their construction.

We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold corresponds to an orbit of the holonomy group on the modelling homogeneous space and carries a canonical induced Cartan geometry. The result can therefore be understood as a `curved orbit decomposition'. The theory is then applied to the study of several invariant overdetermined differential equations in projective, conformal and CR-geometry. This makes use of an equivalent description of solutions to these equations as parallel sections of a tractor bundle. In projective geometry we study a third order differential equation that governs the existence of a compatible Einstein metric. In CR-geometry we discuss an invariant equation that governs the existence of a compatible K\"{a}hler-Einstein metric., Comment: v2: major revision; 30 pages v3: final version to appear in Duke Math. J

In a previous work with Thorbergsson, it was proved that a simple closed curve in the real projective plane $\mathbf{P}^{2}$ that is not null-homotopic has at least three sextactic points. This assertion was conjectured by Gerrit Bol. That proof used an axiomatic approach called ‘intrinsic conic system’. The purpose of this paper is to give a more elementary proof.

Affine differential geometry developed by Blaschke and his school [B] has been reorganized in the last several years as geometry of affine immersions. An immersion f of an n-dimensional manifold M with an affine connection ∇ into an (n + 1)-dimensional manifold Ḿwith an affine connection ∇ is called an affine immersion if there is a transversal vector field ξ such that ∇xf*(Y) = f*(∇xY) + h(X,Y)ξ holds for any vector fields X, Y on Mn. When f: Mn→ Rn+1 is a nondegenerate hypersurface, there is a uniquely determined transversal vector field ξ, called the affine normal field, an essential starting point in classical affine differential geometry. The new point of view allows us to relax the non-degeneracy condition and gives us more freedom in choosing ξ; what this new viewpoint can accomplish in relating affine differential geometry to Riemannian geometry and projective differential geometry can be seen from [NP1], [NP2], [NS] and others. For the definitions and basic formulas on affine immersions, centroaffine immersions, conormal (or dual) maps, projective flatness, etc., the reader is referred to [NP1]. These notions will be generalized to codimension 2 in this paper.

The generic singularities and bifurcations are classified for one-parameter families of curves with frames in the space forms E n + 1 , S n + 1 , H n + 1 . Two kinds of frames are considered; adapted frames and osculating frames. In particular, we give the classification results on the singularities of envelopes associated to framed curves. The associated envelopes and their singularities are classified. characterised in term of geometric invariants of framed curves. We apply to the global problem of framed curves and to the extension problem of surfaces with boundaries in three space. generalising the results obtained in Ishikawa (2010) [12] .

We solve the problem on flat extensions of a generic surface with boundary in Euclidean 3-space, relating it to the singularity theory of the envelope generated by the boundary. We give related results on Legendre surfaces with boundaries via projective duality and observe the duality on boundary singularities. Moreover we give formulae related to remote singularities of the boundary-envelope., Several typos are corrected (2010. Jan.)

Using the Plucker map between grassmannians, we study basic aspects of classic grassmannian geometries. For `hyperbolic' grassmannian geometries, we prove some facts (for instance, that the Plucker map is a minimal isometric embedding) that were previously known in the `elliptic' case., 12 pages. 2010 edition

A simple closed curve $\gamma$ in the real projective plane $P^2$ is called anti-convex if for each point $p$ on the curve, there exists a line which is transversal to the curve and meets the curve only at $p$. We shall prove the relation $i(\gamma)-2\delta(\gamma)=3$ for anti-convex curves, where $i(\gamma)$ is the number of independent (true) inflection points and $\delta(\gamma)$ the number of independent double tangents. This formula is a refinement of the classical M\"obius theorem. We shall also show that there are three inflection points on a given anti-convex curve such that the tangent lines at these three inflection points cross the curve only once. Our approach is axiomatic and can be applied in other situations. For example, we prove similar results for curves of constant width as a corollary., Comment: 28pages, 20 figures

We prove that a connected properly immersed minimal surface in Euclidean 3-space with infinite symmetry group whose intersection with a ball of radius R is less than 2\piR^2 is a plane, a catenoid or a Scherk singly-periodic minimal surface. In particular, we prove that the only periodic minimal desingularization of a pair of intersecting planes is Scherk's singly-periodic minimal surface., 33 pages; 1 figure