Let $K$ be the sectional curvature of a compact submanifold $M$ of the Cayley projective plane $CaP^{2}$. In this paper, we prove that the compact totally complex submanifold $M$ of complex dimension 2 in $CaP^{2}$ satisfying $K \gt (1/8)$ is totally geodesic and $M=CP^{2}$.

In this paper, we investigate a real hypersurface of a complex projective space $CP^{n}$ in terms of the Jacobi operators. We give a local stmcture theorem of a real hypersurface of $CP^{n}$ satisisfying $R_{\xi}=k(I-\eta\otimes\xi)$, where $ R_{\xi}=R(\cdot, \xi)\xi$ the Jacobi operator with respect to $\xi$ and $k$ is a function. Further, we classify real hypersurfaces of $CP^{n}$ satisfying $\phi R_{\xi}=R_{\xi}\phi$ under the condition that $ A\xi$ is a principal curvature vector. Also, we show that a complex projective space does not admit a locally symmetric real hypersurface.

In this paper, we define the locally nilpotent space and condition $(T^{**})$, and study their properties. Furthermore, we find the vanishing condition of the Wall invariant of the space satisfying the condition $(T^{**})$ and locally nilpotent space.

On a real hypersurface of quatemionic projective space $QP^{m}$ we study the following condition: $\mathfrak{S}(R(X, Y)SZ)=0$ where $\mathfrak{S}$ denotes the cyclic sum, $R$, respectively $S$, the curvature tensor, respectively the Ricci tensor, of the real hypersurface and $X,Y\in \mathscr{D}$, $Z\in \mathscr{D}^{\perp}, \mathscr{D}$ and $\mathscr{D}^{\perp}$ being certain distributions on the real hypersurface. We prove that such a real hypersurface must be Einstein.

Relations are studied concerning lightness of a mapping between continua and lightness of the two induced mappings: between hyperspaces of closed subsets and between hyperspaces of subcontinua.

We prove existence of global solutions to a semilinear massless Dirac system with small initial data. We study solutions in generalised Sobolev spaces suggested by S. Klainerman. Our approach is based on using conservation law of charge together with Sobolev type weighted estimates for the spinor field. Our result seems to be sharp in a view of blowing-up results obtained by F. John (see [7]). We also study decay properties of the spinor field. With similar methods we prove global existence for a nonlinear wave equation in three space dimension. The same equation was studied by T. Sideris [14] and H. Takamura [15]. They proved global existence for spherically symmetrical initial data. In this work we remove this condition on the initial data.

We characterize the real hypersurfaces of a non-flat quatemionic space form with constant quatemionic sectional curvature as the ones which have constant totally real sectional curvature.

The purpose of this paper is to characterize a class of non-degenerate ruled surfaces in $R_{1}^{3}$, which are said to be null scrolls, satisfying the condition $\Delta\xi=A\xi$, where $\xi$ denote their Gauss maps and $A\in gl(3, R)$.