In this paper we prove that if the $r$-th tensor power of the tangent bundle of a smooth projective variety $X$ contains the determinant of an ample vector bundle of rank at least $r$, then $X$ is isomorphic either to a projective space or to a smooth quadric hypersurface. Our result generalizes Mori's, Wahl's, Andreatta-Wiśniewski's and Araujo-Druel-Kovács's characterizations of projective spaces and hyperquadrics.

In this paper we present a topological way of building a compactification of a symmetric space from a compactification of a Weyl Chamber. We also present a way to achieve compactifications by means of generalized Busemann functions.

Let $\Delta \subset \mathbb{R}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_{\Delta}$ and a very ample $(\mathbb{C}×)^n$-equivariant line bundle $L_{\Delta}$ on $X_{\Delta}$ associated with $\Delta$. In the present paper, we give a necessary and sufficient condition for Chow semistability of $( X_{\Delta}, {L^i}_{\Delta})$ for a maximal torus action. We then see that asymptotic (relative) Chow semistability implies (relative) K-semistability for toric degenerations, which is proved by Ross and Thomas.

We consider the problem of whether a given spacetime admits a foliation by $(n−1)$-umbilical spacelike hypersurfaces. We introduce the notion of a timelike closed partially conformal vector field in a spacetime and show that the existence of a vector field of this kind guarantees in turn the existence of that foliation. We then construct explicit examples of families of $(n−1)$-umbilical spacelike hypersurfaces in the de Sitter space. Imposing the further condition of having constant $r$-th mean curvature, we give the complete description of any leaf of a foliation of the de Sitter space by these hypersurfaces. Finally, in a spacetime foliated by $(n−1)$-umbilical spacelike hypersurfaces we characterize the immersed spacelike hypersurfaces which are $(n−1)$-umbilical.

This paper studies the homotopical minimal period of self-maps on flat manifolds with cyclic holonomies. We give some necessary conditions on the self-maps on flat manifolds to guarantee that their homotopical minimal periods are infinite. Furthermore, a kind of density of homotopical minimal periods in the natural number set is considered.

In this paper, we put the issue of dynamic equivalence of control systems in the context of pullbacks of coframings on infinite jet bundles over the state manifolds. While much attention has been given to differentially flat systems, i.e., systems dynamically equivalent to linear control systems, the advantage of this approach is that it allowed us to consider control affine systems as well. Through this context we are able to classify all control affine systems of three states and two controls under dynamic equivalence of the type $(x,u)\mapsto y(x,u)$.

Let $k$ be any field, $G$ be a finite group acting on the rational function field $k(x_g : g \in G)$ by $h \cdot x_g = x_{hg}$ for any $h, g \in G$. Define $k(G) = k(x_g : g \in G)^G$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is known that, if $\mathbb{C}(G)$ is rational over $\mathbb{C}$, then $B_0(G) = 0$ where $B_0(G)$ is the unramified Brauer group of $\mathbb{C}(G)$ over $\mathbb{C}$. Bogomolov showed that, if $G$ is a $p$-group of order $p^5$, then $B_0(G) = 0$. This result was disproved by Moravec for $p = 3, 5, 7$ by computer calculations. We will prove the following theorem. Theorem. Let $p$ be any odd prime number, $G$ be a group of order $p^5$. Then $B_0(G) \neq 0$ if and only if $G$ belongs to the isoclinism family ${\Phi}_{10}$ in R. James's classification of groups of order $p^5$.

In this paper we give a new proof (along the line of the original proof) for the almost isometry theorem in Alexandrov geometry with curvature bounded below in Yu. Burago, M. Gromov, and G. Perel’man, A. D. Alexandrov spaces with curvature bounded below, Russian Math. Surveys, 47:2 (1992), pp. 1–58. The motivation of the new proof is that we find that Lemma 9.11 in A. D. Alexandrov spaces with curvature bounded below is incorrect (see Example 1.3 below), while this lemma is a crucial step in the original proof.