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In this paper, based on the classical K. Yano's formula, we first establish an optimal integral inequality for compact Lagrangian submanifolds in the complex space forms, which involves the Ricci curvature in the direction $J\vec{H}$ and the norm of the covariant differentiation of the second fundamental form $h$, where $J$ is the almost complex structure and $\vec{H}$ is the mean curvature vector field. Second and analogously, for compact Legendrian submanifolds in the Sasakian space forms with Sasakian structure $(\varphi,\xi,\eta,g)$, we also establish an optimal integral inequality involving the Ricci curvature in the direction $\varphi\vec{H}$ and the norm of the modified covariant differentiation of the second fundamental form. The integral inequality is optimal in the sense that all submanifolds attaining the equality are completely classified. As direct consequences, we obtain new and global characterizations for the Whitney spheres in complex space forms as well as the contact Whitney spheres in Sasakian space forms. Finally, we show that, just as the Whitney spheres in complex space forms, the contact Whitney spheres in Sasakian space forms are locally conformally flat manifolds with sectional curvatures non-constant., Comment: 12 pages

In this paper we consider minimal Lagrangian submanifolds in $n$-dimensional complex space forms. More precisely, we study such submanifolds which, endowed with the induced metrics, write as a Riemannian product of two Riemannian manifolds, each having constant sectional curvature. As the main result, we give a complete classification of these submanifolds.

In this paper, we study locally strongly convex Tchebychev hypersurfaces, namely the {\it centroaffine totally umbilical hypersurfaces}, in the $(n+1)$-dimensional affine space $\mathbb{R}^{n+1}$. We first make an ordinary-looking observation that such hypersurfaces are characterized by having a Riemannian structure admitting a canonically defined closed conformal vector field. Then, by taking the advantage of properties about Riemannian manifolds with closed conformal vector fields, we show that the ellipsoids are the only centroaffine Tchebychev hyperovaloids. This solves the longstanding problem of trying to generalize the classical theorem of Blaschke and Deicke on affine hyperspheres in equiaffine differential geometry to that in centroaffine differential geometry., Comment: 14 pages

In this paper, extending our previous joint work (Hu et al., Math Nachr 291:343--373, 2018), we initiate the study of Hopf hypersurfaces in the homogeneous NK (nearly K\"ahler) manifold $\mathbf{S}^3\times\mathbf{S}^3$. First, we show that any Hopf hypersurface of the homogeneous NK $\mathbf{S}^3\times\mathbf{S}^3$ does not admit two distinct principal curvatures. Then, for the important class of Hopf hypersurfaces with three distinct principal curvatures, we establish a complete classification under the additional condition that their holomorphic distributions $\{U\}^\perp$ are preserved by the almost product structure $P$ of the homogeneous NK $\mathbf{S}^3\times\mathbf{S}^3$., Comment: 26 pages

Each hypersurface of a nearly K\"ahler manifold is naturally equipped with two tensor fields of $(1,1)$-type, namely the shape operator $A$ and the induced almost contact structure $\phi$. In this paper, we show that, in the homogeneous NK $\mathbb{S}^6$ a hypersurface satisfies the condition $A\phi+\phi A=0$ if and only if it is totally geodesic; moreover, similar as for the non-flat complex space forms, the homogeneous nearly K\"ahler manifold $\mathbb{S}^3\times\mathbb{S}^3$ does not admit a hypersurface that satisfies the condition $A\phi+\phi A=0$., Comment: 14 pages. All comments are welcome

In this paper, we study Lagrangian submanifolds of the homogeneous nearly K\"ahler $6$-dimensional unit sphere $\mathbb{S}^6(1)$. As the main result, we derive a Simons' type integral inequality in terms of the second fundamental form for compact Lagrangian submanifolds of $\mathbb{S}^6(1)$. Moreover, we show that the equality sign occurs if and only if the Lagrangian submanifold is either the totally geodesic $\mathbb{S}^3(1)$ or the Dillen-Verstraelen-Vrancken's Berger sphere $S^3$ discribed in J Math Soc Japan, 42: 565-584, 1990., Comment: 12 pages. In this updated version we include a new rigidity theorem i.e. Theorem 4.1. All comments are welcome

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space $\mathbb{R}^{n+1}$ which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvatures. As the main result, a complete classification of such affine hyperspheres is established. Moreover, as direct consequences, affine hyperspheres of dimensions 3 and 4 with parallel Ricci tensor are also classified., Comment: 27 pages. This article has been accepted for publication in SCIENCE CHINA Mathematics

In this paper, we establish a general inequality for locally strongly convex centroaffine hypersurfaces in $\mathbb{R}^{n+1}$ involving the norm of the covariant derivatives of both the difference tensor $K$ and the Tchebychev vector field $T$. Our result is optimal in that, applying our recent classification for locally strongly convex centroaffine hypersurfaces with parallel cubic form in [4], we can completely classify the hypersurfaces which realize the equality case of the inequality., Comment: 11 pages

The equivariant CR minimal immersions from the round $3$-sphere $S^3$ into the complex projective space $\mathbb CP^n$ have been classified by the third author explicitly (J London Math Soc 68: 223-240, 2003). In this paper, by employing the equivariant condition which implies that the induced metric is left-invariant, and that all geometric properties of $S^3={\rm SU}(2)$ endowed with a left-invariant metric can be expressed in terms of the structure constants of the Lie algebra $\mathfrak{su}(2)$, we establish an extended classification theorem for equivariant CR minimal immersions from the $3$-sphere $S^3$ into $\mathbb CP^n$ without the assumption of constant sectional curvatures., Comment: 23 pages;This article has been accepted for publication in Annals of Global Analysis and Geometry

In this paper, we study locally strongly convex centroaffine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the centroaffine metric. As the main result, we obtain a complete classification of such centroaffine hypersurfaces. The result of this paper is a centroaffine version of the complete classification of locally strongly convex equiaffine hypersurfaces with parallel cubic form due to Hu, Li and Vrancken [12]., Comment: 44 pages

In this paper, we show that isotropic Lagrangian submanifolds in a $6$-dimensional strict nearly K\"ahler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the $J$-isotropic Lagrangian submanifolds in the homogeneous nearly K\"ahler $\mathbb{S}^3\times \mathbb{S}^3$ is also obtained. Here, a Lagrangian submanifold is called $J$-isotropic, if there exists a function $\lambda$, such that $g((\nabla h)(v,v,v),Jv)=\lambda$ holds for all unit tangent vector $v$., Comment: 18 pages. This article has been accepted for publication in SCIENCE CHINA Mathematics