We consider a Riemannian manifold $M$ with an $f$-structure. With some additional properties such a manifold is called a $\mathscr{K}, \mathscr{C} or \mathscr{S}$-manifold. The considered structures determine a Riemannian foliation, whose leaf closures form a singular Riemannian foliation. We give conditions under which the foliation of the principal stratum is again associated to a structure of the type we consider. The manifold can be partitioned into strata on which the leaf closures are given by toroidal fiber bundles. This theorem is a topological generalization of the classical Boothby-Wang theorem for the contact manifolds.

We show that there is no $CR$ submanifold with semi-flat normal connection and with recurrent Ricci tensor in a complex space form of nonzero constant holomorphic sectional curvature, if the dimension of its holomorphic distribution is greater than 2.

We prove a sphere theorem for manifolds referred to spherical warped product models and obtain the optimal result.

$\mathrm{Cliff}(X)$ be the Clifford index of a curve $X$ and $L$ be a base-point-free line bundle on $X$ which satisfies $\mathrm{Cliff}(L) = \mathrm{Cliff}(X) + k (k \geq 3)$. We determine sufficient conditions for the $|L| = g^{r}_{d}$ being simple (i.e. birationally very ample).

Let $\mathrm{Cliff}(X)$ be the Clifford index of a curve. We determine necessary conditions for very ample line bundles of degree $\mathrm{deg}(L) = 2g - 2h^{1}-\mathrm{Cliff}(X) - k (k=2,3,4)$ being not normally generated.

We consider the wave equations with perturbation of the first order differential operators, the coeffcients of which depend on both the space and time variables. Under suitable conditions on the coeffcients, we show the existence of the scattering operator. The main tool used is space-time energy estimates of solutions.

We give kinematic formulae for integral invariants of degree 2 for hypersurfaces in two point homogeneous spaces explicitly. The discussion here we use is a certain generalization of the transfer principle in integral geometry.

We describe a method of manufacturing harmonic maps between semi-Riemannian spheres out of those in Riemannian geometry. After normalization, the resulting maps give examples of helical geodesic immersions in semi-Riemannian geometry.