Let $\Phi$ be an Hermitian quadratic form, of maximal rank and index $(n,1)$, defined over a complex $(n+1)$ vector space $V$. Consider the real hyperquadric defined in the complex projective space $P^{n}V$ by $$Q=\{\left[\varsigma \right]\in P^{n}V,\Phi \left(\varsigma \right)=0\}.$$ Let $G$ be the subgroup of the special linear group which leaves $Q$ invariant and $D$ the $\left(2n\right)-$ distribution defined by the Cauchy Riemann structure induced over $Q$. We study the real regular curves of constant type in $Q$, tangent to $D$, finding a complete system of analytic invariants for two curves to be locally equivalent...

We introduce an infinite sequence of higher order Schwarzian derivatives closely related to the theory of monotone matrix functions. We generalize the classical Koebe lemma to maps with positive Schwarzian derivatives up to some order, obtaining control over derivatives of high order. For a large class of multimodal interval maps we show that all inverse branches of first return maps to sufficiently small neighbourhoods of critical values have their higher order Schwarzian derivatives positive up...

We prove that any compact Kähler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact Kähler manifold. This shows that many complex manifolds admit no or few holomorphic Cartan geometries.

Une métrique riemannienne holomorphe sur une variété complexe $M$ est une section holomorphe $q$ du fibré $S^2\left(T^{*}M\right)$ des formes quadratiques complexes sur l’espace tangent holomorphe à $M$ telle que, en tout point $m$ de $M$, la forme quadratique complexe $q\left(m\right)$ est non dégénérée (de rang maximal, égal à la dimension complexe de $M$). Il s’agit de l’analogue, dans le contexte holomorphe, d’une métrique riemannienne (réelle). Contrairement au cas réel, l’existence d’une telle métrique sur une variété complexe compacte n’est...

The concept of homogeneity, which picks out sprays from the general run of systems of second-order ordinary differential equations in the geometrical theory of such equations, is generalized so as to apply to equations of higher order. Certain properties of the geometric concomitants of a spray are shown to continue to hold for higher-order systems. Third-order equations play a special role, because a strong form of homogeneity may apply to them. The key example of a single third-order equation...