In this paper, we generalize the Gauduchon metrics on a compact complex manifold and define the ${\gamma }_k$ functions on the space of its hermitian metrics.

$\mathrm{𝑆𝑒𝑐𝑜𝑛𝑑}-\mathrm{𝑜𝑟𝑑𝑒𝑟𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐𝐿𝑜𝑟𝑒𝑛𝑡𝑧𝑖𝑎𝑛𝑠𝑝𝑎𝑐𝑒𝑠}$, that is to say, Lorentzian manifolds with vanishing second derivative $\nabla \nabla R\equiv 0$ of the curvature tensor $R$, are characterized by several geometric properties, and explicitly presented. Locally, they are a product $M=M_1\times M_2$ where each factor is uniquely determined as follows: $M_2$ is a Riemannian symmetric space and $M_1$ is either a constant-curvature Lorentzian space or a definite type of plane wave generalizing the Cahen–Wallach family. In the proper case (i.e., $\nabla R\ne 0$ at some point), the curvature tensor turns out to...

On étudie la structure naturelle d’algèbre de Lie de l’espace des sections de classe $C_k$ d’un fibré localement trivial dont la fibre-type est une algèbre de Lie $L$; on décrit, en particulier, ses dérivations et ses automorphismes. On détermine les algèbres de Lie $L$ pour lesquelles cette structure caractérise la structure différentiable de la base du fibré.