In this paper we prove that a six-dimensional quasi Kähler manifold of constant sectional curvature is a nearly Kähler manifold of positive sectional curvature or a flat Kähler manifold.

Examples are presented showing that some statements (and the main results) of the paper [12] ``On unicoherence at subcontinua'' by Zhou Youcheng are not true. For some other assertions from that paper, which are correctly stated but whose proofs contain essential gaps, the proper arguments are supplied.

In this note, we shall calculate some homogeneous polynomials explicitly based on B. Runge’s conjecture in [4] p. 203 and give the explicit structure of the graded ring $A(\Gamma_{2}(1,2))$ of Siegel modular forms of genus two belonging to the discrete subgroup of $\Gamma_{2}(1,2)$ of $Sp(2, R)$.

Lagrangian $H$-umbilical submanifolds introduced in $[1, 2]$ can be regarded as the simplest Lagrangian submanifolds in Kaehler manifolds next to totally geodesic ones. It was proved in [1] that Lagrangian $H$-umbilical submanifolds of dimension $\geq 3$ in complex Euclidean spaces are complex extensors, Lagrangian pseudo-spheres, and flat Lagrangian $H$-umbilical submanifolds. Lagrangian $H$-umbilical submanifolds of dimension $\geq 3$ in non-flat complex space forms are classified in [2]. In this paper we deal with the remaining case; namely, non-totally geodesic Lagrangian $H$-umbilical surfaces in complex space forms. Such Lagrangian surfaces are characterized by a very simple property; namely, $JH$ is an eigenvector of the shape operator $A_{H}, where $H$ is the mean curvature vector field. The main operator of this paper is to determine both the intrinsic and the extrinsic structures of Lagrangian $H$-umbilical surfaces.

We study the topological structure of the closed orientable 3-manifolds obtained by Dehn surgeries along certain links, first considered by Takahashi in [23]. The interest about such manifolds arises from the fact that they include well-known families of 3-manifolds, previously studied by several authors, as the Fibonacci manifolds [7], [10], [11], the Fractional Fibonacci manifolds [14], and the Sieradski manifolds [5], [6], respectively. Our main result states that the Takahashi manifolds are 2-fold coverings of the 3-sphere branched along the closures of specified 3-string braids. We also describe many of the above-mentioned manifolds as n-fold cyclic branched coverings of the 3-sphere.

We prove that the product space $X\times Y$ is collectionwise normal if and only if $X\times Y$ is normal in the following cases; (1) $X$ is a collectionwise normal $\Sigma$-space and $Y$ is a collectionwise normal first countable $P$-space, (2) $X$ is the closed image of a normal M-space and $Y$ is a collectionwise normal first countable $P$-space, (3) $X$ is the closed image of a paracompact M-space and $Y$ is a collectionwise normal $P$-space. In particular, (2) and (3) essentially generalize K. Chiba’s theorems [3].

In this paper, we investigate an isometric immersion of a compact connected Riemannian manifold $M$ into a Euclidean space and a sphere such that every geodesic in $M$ is closed and viewed as a helix (of general order) in the ambient space.