For any closed, oriented, simply connected spin 4-manifold $X$ with $b_{X}^{+}>1$ and even, by Fintushel and Stern [7], differential-topological polynomial invariants with its values in $Z_{2}$ are defined. These invariants are analogues of Donaldson polynomial invariants. In [7], it is proved that if $X=X^{\prime}\# S^{2}\times S^{2}$, then these invariants do not always vanish. But in this paper, it is proved that these invariants vanish for a large class of connected sums.

A regular submanifold in a Euclidean space $R^{N}$ is called a submanifold with simple geodesics if all its geodesics have constant Frenet curvatures in $R^{N}$. A submanifold with congruent simple geodesics is called helical. We prove that a compact surface with simple geodesics is either a rational torus, or the image of the unit sphere under a polynomial map $F:R^{3}\rightarrow R^{N}$ of the special structure. As a corollary, a compact surface $F^{2}\subset R^{N}$ is helical if $F^{2}=\Phi(S^{2})$, where $\Phi=(a_{1}\Phi_{1}, \ldots, a_{m}\Phi_{m})$ and $\Phi_{j}$ the $i-$th eigenmap of the Laplacian of $S^{2}$.

Let $X$ be a metric continuum, let $2^{X}$ be the hyperspaces of all the nonempty closed subsets of $X$ and let $C(X)$ be the hyperspace of subcontinua of $X$. In this paper we prove: THEOREM 1. If $\mathscr{H}$ is a O-dimensional subset of $2^{X}$, then $2^{X}-\mathscr{H}$ is connected. THEOREM 2. If $\mathscr{H}$ is a closed O-dimensional subset of $C(X)$ such that $C(X)-\{A\}$ is arcwise connected for each $A\in \mathscr{H}$,then $C(X)-\mathscr{H}$ is arcwise connected. Theorem 2 answers a question by Sam B. Nadler, Jr.

We prove that $\mathscr{K}(Q)$ is a topological group and characterize $\mathscr{K}(Q)$ as a first-category, zero-dimensional, separable, metrizable space of which every non-empty clopen subset is $\Pi_{1}^{1}$-complete. In particular we answer a question of Fujita and Taniyama ([5]). With the additional assumption of Analytic Determinacy it was proved in [5] that $\mathscr{K}(Q)$ is a homogeneous space.

Let $A$ denote the class of analytic functions $f(z)$ defined in the unit disc satisfying the condition $f(O)=f^{\prime}(0)-1=0$. Let $\overline{S}^{*}(\beta, \gamma)$ be the class of strongly starlike functions of order $\beta$ and type $\gamma$, and let $\overline{C}(\beta, \gamma)$ denote the class of strongly convex functions of order $\beta$ and type $\gamma$. Certain new classes $\overline{S}_{\alpha}^{*}(\beta, \gamma)$ and $\overline{C}_{\alpha}(\beta, \gamma)$ are introduced by virtue of Ruscheweyh derivative and some properties of $\overline{S}_{\alpha}^{*}(\beta, \gamma)$ and $\overline{C}_{\alpha}(\beta, \gamma)$ are discussed.

Let $(F, \leq)$ be an ordered field. With respect to the order topology, $F$ is a Tychonoff uniform space. $F$ is metrizable if and only if there is a countable set $\{b_{1}, \ldots, b_{n}, \ldots\}$ of positive elements of $F$ such that if $b$ is any positive element of $F$, there exists $n\geq 1$ such that $0 \lt b_{n} \lt b$. If $F$ is denumerable or Archimedean, then this metrizability condition is satisfied. For each uncountable cardinal number $\aleph$, there exist ordered fields, $F_{1}$ and $F_{2}$, each of cardinality $\aleph$, such that the order topology on $F_{1}$ (resp., $F_{2}$) is (resp., is not) metrizable.

In this paper we study the curvature pinching property for minimal surfaces in complex Grassmann manifolds and obtain some results.