Nous présentons plusieurs résultats de rigidité concernant les flots d’Anosov admettant transversalement des structures symplectiques réelles ou complexes de dimension $5$.

We introduce the new notion of pseudo-$𝔻$-parallel real hypersurfaces in a complex projective space as real hypersurfaces satisfying a condition about the covariant derivative of the structure Jacobi operator in any direction of the maximal holomorphic distribution. This condition generalizes parallelness of the structure Jacobi operator. We classify this type of real hypersurfaces.

We characterize homogeneous real hypersurfaces of types (A₀), (A₁) and (B) in a complex projective space or a complex hyperbolic space.

In this paper we prove a non-existence of real hypersurfaces in complex hyperbolic two-plane Grassmannians SU2.m/S(U2·Um), m≥3, whose structure tensors {ɸi}i=1,2,3 commute with the shape operator.

We give a classification of Hopf real hypersurfaces in complex hyperbolic two-plane Grassmannians ${\mathrm{SU}}_{2,m}/S(U_2·U_m)$ with commuting conditions between the restricted normal Jacobi operator ${\overline{R}}_N\phi$ and the shape operator $A$ (or the Ricci tensor $S$).

This paper consists of two parts. In the first, we find some geometric conditions derived from the local symmetry of the inverse image by the Hopf fibration of a real hypersurface $M$ in complex space form $M_m\left(4ϵ\right)$. In the second, we give a complete classification of real hypersurfaces in $M_m\left(4ϵ\right)$ which satisfy the above geometric facts.

We prove the non-existence of real hypersurfaces in complex two-plane Grassmannians whose normal Jacobi operator is of Codazzi type.

Lee, Kim and Suh (2012) gave a characterization for real hypersurfaces $M$ of Type $\left(\mathrm{A}\right)$ in complex two plane Grassmannians $G_2\left({ℂ}^{m+2}\right)$ with a commuting condition between the shape operator $A$ and the structure tensors $\phi$ and ${\phi }_1$ for $M$ in $G_2\left({ℂ}^{m+2}\right)$. Motivated by this geometrical notion, in this paper we consider a new commuting condition in relation to the shape operator $A$ and a new operator $\phi {\phi }_1$ induced by two structure tensors $\phi$ and ${\phi }_1$. That is, this commuting shape operator is given by $\phi {\phi }_{1}A=A\phi {\phi }_1$. Using this condition, we prove that...

In this paper, first we introduce a new notion of commuting condition that $\phi {\phi }_{1}A=A{\phi }_1\phi$ between the shape operator $A$ and the structure tensors $\phi$ and ${\phi }_1$ for real hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$. Suprisingly, real hypersurfaces of type $\left(A\right)$, that is, a tube over a totally geodesic $G_2\left({ℂ}^{m+1}\right)$ in complex two plane Grassmannians $G_2\left({ℂ}^{m+2}\right)$ satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in $G_2\left({ℂ}^{m+2}\right)$ satisfying the commuting condition. Finally we get a characterization of Type $\left(A\right)$ in terms of such commuting...

In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space form $M_m\left(c\right)$, $c\ne 0$ as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional curvature given in [11].

We characterize real hypersurfaces with constant holomorphic sectional curvature of a non flat complex space form as the ones which have constant totally real sectional curvature.

We show, using a direct variational approach, that the second boundary value problem for the Monge-Ampère equation in ${ℝ}^n$ with exponential non-linearity and target a convex body $P$ is solvable iff $0$ is the barycenter of $P.$ Combined with some toric geometry this confirms, in particular, the (generalized) Yau-Tian-Donaldson conjecture for toric log Fano varieties $(X,\Delta )$ saying that $(X,\Delta )$ admits a (singular) Kähler-Einstein metric iff it is K-stable in the algebro-geometric sense. We thus obtain a new proof and...

Soit $K_0\in ]-1,0[$; chaque métrique complète à courbure $K_0$ sur la sphère à $N\ge 1$ trous admet une unique réalisation comme métrique induite sur une surface plongée dans $H^3$ dont le bord à l’infini est une réunion disjointe de cercles. De manière duale, chaque métrique complète à courbure ${\tilde{K}}_0\in ]-\infty ,0[$ sans géodésique fermée de longueur $L\le 2\pi$ se réalise de manière unique comme troisième forme fondamentale d’une surface plongée dont le bord à l’infini est une réunion de cercles.