There exist exactly four homomorphisms $\varphi$ from the pseudo-orthogonal group of index one $G=O(n,1,ℝ)$ into the group of real numbers ${ℝ}_0.$ Thus we have four $G$-spaces of $\varphi$-scalars $(ℝ,G,h_{\varphi })$ in the geometry of the group $G.$ The group $G$ operates also on the sphere $S^{n-2}$ forming a $G$-space of isotropic directions $(S^{n-2},G,*).$ In this note, we have solved the functional equation $F(A*q_1,A*q_2,\cdots ,A*q_m)=\varphi \left(A\right)·F(q_1,q_2,\cdots ,q_m)$ for given independent points $q_1,q_2,\cdots ,q_m\in S^{n-2}$ with $1\le m\le n$ and an arbitrary matrix $A\in G$ considering each of all four homomorphisms. Thereby we have determined all equivariant mappings $F:{\left(S^{n-2}\right)}^m\to ℝ.$

We show how the ad hoc prescriptions appearing in 2001 for the Lie derivative of Lorentz tensors are a direct consequence of the Kosmann lift defined earlier, in a much more general setting encompassing older results of Y. Kosmann about Lie derivatives of spinors.

We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups $\mathrm{GA}\left(2\right)=\mathrm{GL}(2,ℝ)⋉{ℝ}^2$ and $\mathrm{GA}\left(3\right)=\mathrm{GL}(3,ℝ)⋉{ℝ}^3$, respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective...

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...

We present a generalization of the concept of semiholonomic jets within the framework of higher order prolongations of a fibred manifold. In this respect, a compilation of our 2-fibred manifold approach with the methods of natural operators theory is used.

The problem of invariant output tracking is considered: given a control system admitting a symmetry group $G$, design a feedback such that the closed-loop system tracks a desired output reference and is invariant under the action of $G$. Invariant output errors are defined as a set of scalar invariants of $G$; they are calculated with the Cartan moving frame method. It is shown that standard tracking methods based on input-output linearization can be applied to these invariant errors to yield the required...

The problem of invariant output tracking is considered: given a control system admitting a symmetry group G, design a feedback such that the closed-loop system tracks a desired output reference and is invariant under the action of G. Invariant output errors are defined as a set of scalar invariants of G; they are calculated with the Cartan moving frame method. It is shown that standard tracking methods based on input-output linearization can be applied to these invariant errors to yield the...

In the tangent plane at any point of a surface in the four-dimensional Euclidean space we consider an invariant linear map ofWeingarten-type and find a geometrically determined moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of surfaces...