Jacobi fields of the Tanaka-Webster connection on Sasakian manifolds

We build a variational theory of geodesics of the Tanaka-Webster connection on a strictly pseudoconvex CR manifold.Comment: 52 page

Subelliptic harmonic morphisms

We study subelliptic harmonic morphisms i.e. smooth maps $\phi: \Omega \to \tilde\Omega$ among domains $\Omega \subset \mathbb{R}^n$ and $\tilde\Omega \subset \mathbb{R}^M$ endowed with Hörmander systems of vector fields $X$ and $Y$, that pull back local solutions to $H_Y v = 0$ into local solutions to $H_X u = 0$, where $H_X$ and $H_Y$ are Hörmander operators. We show that any subelliptic harmonic morphism is an open mapping. Using a subelliptic version of the Fuglede-Ishihara theorem (due to E. Barletta [5]) we show that given a strictly pseudoconvex CR manifold $M$ and a Riemannian manifold $N$ for any heat equation morphism $\Psi: M \times (0, \infty) \to N \times (0, \infty)$ of the form $\Psi(x,t) = ( \phi (x), h(t))$ the map $\phi : M \to N$ is a subelliptic harmonic morphism

Applications et Morphismes Harmoniques

DoctoralCes leçons constituent une exposition precise, avec des calculs explicites, d elements introductifs a la theorie des applications harmoniques entre varietes Riemanniennes. On etablit les formules de la premiere et de la seconde variation de l'energie de Dirichlet et on montre certaines de leurs consequences geometriques comme le theoreme de B. Solomon (cf. [34]) et la theorie de la stabilite pour les applications harmoniques. On demontre aussi un theoreme classique de B. Fuglede et T. Ishihara (cf. [18], [23]) sur les morphismes harmoniques. Les morphismes des equations de la chaleur et les morphismes des noyaux de la chaleur (avec variables separees) sont decrits, dans la theorie des morphismes harmoniques, par un resultat tres beau de E. Loubea

Subelliptic biharmonic maps

We study subelliptic biharmonic maps, i.e. smooth maps from a compact strictly pseudoconvex CR manifold M into a Riemannian manifold N which are critical points of a certain bienergy functional. We show that a map is subelliptic biharmonic if and only if its vertical lift to the (total space of the) canonical circle bundle is a biharmonic map with respect to the Fefferman metric.Comment: 23 page

Levi harmonic maps of contact Riemannian manifolds

We study Levi harmonic maps i.e. $C^\infty$ solutions $f : M \to M^\prime$ to \tau_\mathcal{H} (f) \equiv \mathr{trace}_{g} ( \Pi_\mathcal{H}\beta_f ) = 0, where $(M , \eta, g)$ is an (almost) contact (semi) Riemannian manifold, $M^\prime$ is a (semi) Riemannian manifold, $\beta_f$ is the second fundamental form of $f$, and $\Pi_\mathcal{H} \beta_f$ is the restriction of $\beta_f$ to the Levi distribution $\mathcal{H} = \mathrm{Ker}(\eta )$. Many examples are exhibited e.g. the Hopf vector field on the unit sphere $S^{2n+1}$, immersions of Brieskorn spheres, and the geodesic flow of the tangent sphere bundle over a Riemannnian manifold of constant curvature $1$ are Levi harmonic maps. A CR map $f$ of contact (semi) Riemannian manifolds (with spacelike Reeb fields) is pseudoharmonic if and only if $f$ is Levi harmonic. We give a variational interpretation of Levi harmonicity. Any Levi harmonic morphism is shown to be a Levi harmonic map

International Cooperation for Smart and Sustainable Agriculture

This chapter presents international best practices, realized within Europe, and focuses on cooperation for developing innovation support mechanisms and approaches in the area of smart agriculture. Specific situations are presented and analyzed in detailed regarding the requirements of smart agriculture and the possibilities to implement its percepts. As a consequence, solutions are proposed both in the technical and management domains to help speed up the transition from classical agriculture techniques to technology infused approaches, suitable for the current needs of this sector. Also, policy recommendations are developed based on the scientific findings in alignment with the evolution of the competitive pressures