Narrow Results

In this paper, we obtain a sufficient and necessary condition for the existence of symplectic critical surfaces with parallel normalized mean curvature vector in two-dimensional complex space forms. Explicitly, we find that there does not exist any symplectic critical surface with parallel normalized mean curvature vector in two-dimensional complex space forms of nonzero constant holomorphic sectional curvature. And there exists and only exists a two-parameters family of symplectic critical surfaces with parallel normalized mean curvature vector in two-dimensional complex plane, which are rotationally symmetric.

We complete the classification of ruled real hypersurfaces with shape operator of constant norm in nonflat complex space forms by showing the existence of a unique inhomogeneous example in the complex hyperbolic space.

In this paper, we study the interesting open problem of classifying the minimal Lagrangian submanifolds of dimension $n$ in complex space forms with semi-parallel second fundamental form. First, we completely solve the problem in cases $n=2,3,4$. Second, supposing further that the scalar curvature is constant for $n\ge 5$, we also give an answer to the problem by applying the classification theorem of [F. Dillen, H. Li, L. Vrancken and X. Wang, Lagrangian submanifolds in complex projective space with parallel second fundamental form, Pacific J. Math. 255 (2012) 79–115]. Finally, for such Lagrangian submanifolds in the above problem with $n\ge 3$, we establish an inequality in terms of the traceless Ricci tensor, the squared norm of the second fundamental form and the scalar curvature. Moreover, this inequality is optimal in the sense that all the submanifolds attaining the equality are completely determined.

We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Jacobi operator associated to the Weyl conformal curvature tensor of a Riemannian manifold has constant eigenvalues on the bundle of unit tangent vectors and classify such manifolds which are not conformally flat in dimensions congruent to 2 mod 4.

The class of biwarped product manifolds is a generalized class of product manifolds and a special case of multiply warped product manifolds. In this paper, biwarped product submanifolds of the type $N_T{\times }_{{\psi }_1}{N}_{\perp }{\times }_{{\psi }_2}{N}_{𝜃}$ embedded in the complex space forms are studied. Some characterizing inequalities for the existence of such type of submanifolds are derived. Moreover, we also estimate the squared norm of the second fundamental form in terms of the warping function and the slant function. This inequality generalizes the result obtained by Chen in [B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifolds I, Monatsh. Math.133 (2001) 177–195]. By the application of derived inequality, we compute the Dirichlet energies of the warping functions involved. A nontrivial example of these warped product submanifolds is provided.

This paper is devoted to the obtain Chen inequalities containing the Ricci and scalar inequalities on the vertical and the horizontal distributions for semi-slant Riemannian submersions from complex space forms onto Riemannian manifolds. The equality case of the obtained inequalities is discussed. At the end, some geometric consequences are obtained. Moreover, many examples are constructed.

We study m-dimensional real submanifolds of codimension p with (m – 1)-dimensional maximal holomorphic tangent subspace in complex space forms. Consequently, on these manifolds there exists an almost contact structure (F, u, U, g) naturally induced from the ambient space. In this paper we study certain conditions on the almost contact structure and on the second fundamental form of these submanifolds.

We give a simple argument for the nonexistence of Einstein hypersurfaces in the complex space forms CPⁿ and CHⁿ which is valid for all dimensions n ≥ 2. In addition, we survey classification results that are stated in terms of intrinsic geometrical properties of the hypersurface. Many questions that have been settled for Hopf hypersurfaces and/or for dimensions n ≥ 3 still remain open in the general case.

The main purpose of this paper is to demonstrate that there exists no simply connected Hessian manifold (M, ∇, g) of constant Hessian sectional curvature c in the case where (i) ∇ is complete and (ii) c > 0. We achieve the purpose by considering totally geodesic Kähler immersions into a complex space form.