Let $T(A)=A\ltimes D(A)$ be the trivial extension of iterated tilted algebra $A$ of type $\vec{\Delta}$. In this paper, we study the indecomposable $T(A)$-modules belonging to the components of form $Z\vec{\Delta}$, which are called the modules on platform. Our main results are as follows: (1) The number of the modules on platform which have the same dimension vector is equal to or less than the number of simple $A$-modules. (2) The module on platform is uniquely determined by its top and socle. (3) The module on platform is uniquely determined by its Loewy factor and by its socle factor.

We classify cyclic-parallel real hypersurfaces of quaternionic proiective space.

We study certain secondary composition operations that generalise the Toda brackets in order to develop a technique for systematic computation of stable and unstable homotopy pair groups of sphere classes.

One purpose of this paper is to obtain formulae for the shape operators of tubes, whose radius is not necessarily constant, over Riemannian submanifolds. Another purpose of this paper is to define the notion of the natural lift of a distribution on a Riemannian submanifold to a tube over the submanifold and investigate its integrability. Also, we shall construct models of certain kind of Riemannian hypersuface in terms of the formulae for the shape operators of tubes.

In this article, we study the following problem of [5]: Classify all finite type surfaces in a Euclidean 3-space $E^{3}$. A surface $M$ in a Euclidean 3-space is said to be of finite type if each of its coordinate functions is a finite sum of eigenfunctions of the Laplacian operator on $M$ with respect to the induced metric (cf. [1, 2]). Minimal surfaces are the simplest examples of surfaces of finite type, in fact, minimal surfaces are of 1-type. The spheres, minimal surfaces and circular cylinders are the only known examples of surfaces of finite type in $E^{3}$ (cf. [5]). The first author conjectured in [2] that spheres are the only compact finite type surfaces in $E^{3}$. Since then, it was proved step by step and separately that finite type tubes, finite type ruled surfaces, finite type quadrics and finite type cones are surfaces of the only known examples (cf. [2, 6, 7, 10].) Our next natural target for this classification problem is the class of surfaces of revolution. However, this case seems to be much difficult than the other cases mentioned above. We therefore investigate this classification problem for this class and obtain classification theorems for surfaces of revolution which are either of rational or of polynomial kinds (cf. \S 1 for the definitions). As consequence, further supports for the conjecture cited above are obtained.