In this work we give a characterization of the projective invariant pseudometric $P$, introduced by H. Wu, for a particular class of real ${𝐂}^{\mathrm{\infty }}$-manifolds; in view of this result, we study the group of projective transformations for the same class of manifolds and we determine the integrated pseudodistance $p$ of $P$ in open convex regular cones of ${ℝ}^n$, endowed with the characteristic metric.

The aim of this work, which continues Part I with the same title, is to study a class of projective transformations of open, convex, regular cones in ${ℝ}^n$ and to prove a structure theorem for affine transformations of a restricted class of cones; we conclude with a version of the Schwarz Lemma holding for affine transformations.

The projective Finsler metrizability problem deals with the question whether a projective-equivalence class of sprays is the geodesic class of a (locally or globally defined) Finsler function. This paper describes an approach to the problem using an analogue of the multiplier approach to the inverse problem in Lagrangian mechanics.

Fix two points $x,\overline{x}\in S^2$ and two directions (without orientation) $\eta ,\overline{\eta }$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $J\left[\gamma \right]={\int }_0^T\left({}_{\gamma \left(t\right)}(\dot{\gamma }\left(t\right),\dot{\gamma }\left(t\right))+K_{\gamma \left(t\right)}^2{}_{\gamma \left(t\right)}(\dot{\gamma }\left(t\right),\dot{\gamma }\left(t\right))\right)\mathrm{d}t$ along all smooth curves starting from x with direction η and ending in $\overline{x}$ with direction $\overline{\eta }$. Here g is the standard Riemannian metric on S2 and $K_{\gamma }$ is the corresponding geodesic curvature. The interest of this problem comes from mechanics and geometry of vision. It can be formulated as a sub-Riemannian problem on the lens space L(4,1). We...

We study the conditions when locally homogeneous curves in homogeneous spaces admit a natural projective parameter. In particular, we prove that this is always the case for trajectories of homogeneous nilpotent elements in parabolic spaces. On algebraic level this corresponds to the generalization of Morozov–Jacobson theorem to graded semisimple Lie algebras.

Grassmannians of higher order appeared for the first time in a paper of A. Szybiak in the context of the Cartan method of moving frame. In the present paper we consider a special case of higher order Grassmannian, the projective space of second order. We introduce the projective group of second order acting on this space, derive its Maurer-Cartan equations and show that our generalized projective space is a homogeneous space of this group.

Inspired by the results on symmetries of the symplectic Dirac operator, we realize symplectic spinor fields and the symplectic Dirac operator in the framework of (the double cover of) homogeneous projective structure in two real dimensions. The symmetry group of the homogeneous model of the double cover of projective geometry in two real dimensions is $\tilde{}(3,)$.

We consider projectively Anosov flows with differentiable stable and unstable foliations. We characterize the flows on $T^2$ which can be extended on a neighbourhood of $T^2$ into a projectively Anosov flow so that $T^2$ is a compact leaf of the stable foliation. Furthermore, to realize this extension on an arbitrary closed 3-manifold, the topology of this manifold plays an essential role. Thus, we give the classification of projectively Anosov flows on $T^3$. In this case, the only flows on $T^2$ which extend to $T^3$...

First we deduce some general properties of product preserving bundle functors on the category of fibered manifolds. Then we study the prolongation of projectable tangent valued forms with respect to these functors and describe the complete lift of the Frölicher-Nijenhuis bracket. We also present the coordinate formula for composition of semiholonomic jets.

We study systematically the prolongation of second order connections in the sense of C. Ehresmann from a fibered manifold into its vertical bundle determined by a Weil algebra $A$. In certain situations we deduce new properties of the prolongation of first order connections. Our original tool is a general concept of a $B$-field for another Weil algebra $B$ and of its $A$-prolongation.

We prove that the so-called complete lifting of tangent valued forms from a manifold $M$ to an arbitrary Weil bundle over $M$ preserves the Frölicher-Nijenhuis bracket. We also deduce that the complete lifts of connections are torsion-free in the sense of M. Modugno and the second author.