An Einstein-Weyl structure is a natural generalization of an Einstein structure within the framework of conformal geometry. We are interested in considering an Einstein-Weyl structure on a CR manifold. A CR manifold has a conformal structure only on its hyperdistribution. In this paper, on a CR manifold we naturally define an Einstein-Weyl structure closely related to the conformal structure on the hyperdistribution.

We will determine the ring structure of the Hochschild cohomology $HH^{*}(ZQ_{2})$ for the quatemion group $Q_{2}$ of order 8 using the ring isomorphism determined by Siegel and Witherspoon.

We consider two classes of left $R$-modules, $\mathscr{P}$ and $\mathscr{C}$, such that $\mathscr{P}\subset \mathscr{C}$. If the module $M$ has a $\mathscr{P}$-resolution and a $\mathscr{C}$-resolution then for any module $N$ and $n\geq 0$ we define generalized Tate cohomology modules $\widehat{Ext}_{\mathscr{C},\mathscr{P}}^{n}(M, N)$ and show that we get a long exact sequence connecting these modules and the modules $Ext_{\mathscr{C}}^{n}(M, N)$ and $Ext_{\mathscr{P}}^{n}(M, N)$. When $\mathscr{C}$ is the class of Gorenstein projective modules, $\mathscr{P}$ is the class of projective modules and when $M$ has a complete resolution we show that the modules $\widehat{Ext}_{\mathscr{C},\mathscr{P}}^{n}(M, N)$ for $n\geq 1$ are the usual Tate cohomology modules and prove that our exact sequence gives an exact sequence provided by Avramov and Martsinkovsky. Then we show that there is a dual result. We also prove that over Gorenstein rings Tate cohomology $\widehat{Ext}_{R}^{n}(M, N)$ can be computed using either a complete resolution of $M$ or a complete injective resolution of $N$. And so, using our dual result, we obtain Avramov and Martsinkovsky's exact sequence under hypotheses different from theirs.

We will give an explicit description of non-commutative extensions of the additive group scheme (resp. the additive formal group scheme) by the multiplicative group scheme (resp. the multiplicative formal group scheme) over an $F_{p}$-algebra.

We will give an explicit description of extensions of the group scheme of Witt vectors of length $n$ (resp. the formal group scheme of Witt vectotrs of length $n$) by the group scheme (resp. the formal group scheme) which gives a deformation of the additive group scheme to the multiplicative group scheme (resp. the additive formal group scheme to the multiplicative formal group scheme) over an algebra for which all prime numbers except a given prime $p$ are invertible.

In this paper, we treat piecewise Riemannian 2-polyhedra which are combinatorial 2-polyhedra such that each 2-simplex is isometric to a triangle bounded by three smooth curves on some Riemannian 2-manifold. We will introduce the total curvature $C(X)$ of a piecewise Riemannian 2-polyhedron $X$ not only in the compact case but also in the noncompact case, and obtain some generalizations of the Gauss-Bonnet theorem and the Cohn-Vossen theorem. Furthermore, we will show that the difference between $C(X)$ and some value concerning to the topology of $X$ coincides with some expanding growth rate of $X$.

Invariant submanifolds of $(\kappa,\mu)$-manifolds and $(\kappa,\mu)$-space forms are studied.

We know the fact that there are no real hypersurfaces with parallel Ricci tensors in a nonflat complex space form (cf. [5]). In this paper we investigate real hypersurfaces in a nonflat complex space form using some conditions of the Ricci tensor $S$ which are weaker than $\nabla S=0$. We characterize Hopf hypersurfaces of a nonflat complex space form.

Let $\varphi(n)$ be the Euler totient function of $n$. A positive integer $m$ is called a noncototient if the equation $n-\varphi(n)=m$ has no solution in positive integers $n$. The sequence $(2^{k}p)_{k=1}^{\infty}$ which is noncototient for some prime $p$ will be called as Sierpiński's sequence. In this paper we prove some interesting properties of the Sierpiński sequence given in the Theorem 1, 2, 3.

Let $M^{4}$ be a complete noncompact oriented fourdimensional Riemannian manifold satisfying $\delta W^{+}=0$, where $W^{+}$ is the self-dual part of the Weyl curvature tensor. Suppose its scalar curvature is nonnegative and Sobolev's inequality holds. We show that if the $L^{2}$ norm of $W^{+}$ is sufficiently small, then $W^{+}\equiv 0$.