We introduce a generalization of Grassmannians of projective spaces that allows us to consider subspaces of any (possibly infinite) rank as points of the Grassmannian. We show that the spaces that we obtain, carry in a natural way the structure of a twin building.

Let the Riemannian metric of a three-dimensional Kenmotsu manifold be a Yamabe soliton. In this paper, we prove that the Kenmotsu manifold is of constant sectional curvature $-1$ and the Yamabe soliton is expanding with the soliton constant $\lambda=-6$.

We discuss meromorphic solutions of functional equations over non-Archimedean fields, and prove difference analogues of the Clunie lemma, Malmquist-type theorems and the Mokhon'ko theorem.

The purpose of the note is to show the usefulness of a simple criterion of density of a subalgebra $\mathcal A\subset L(X)$ in the strong operator topology for arbitrary real or complex locally convex vector space $X$. After proving the criterion we observe its efficiency obtaining short proofs of three important known density theorems plus a new one.

We characterize metric spaces whose Lipschitz free space is isometric to $\ell_1$. In particular, we show that the Lipschitz free space over an ultrametric space is not isometric to $\ell_1(\Gamma)$ for any set $\Gamma$. We give a lower bound for the Banach-Mazur distance in the finite case.

An integer part $I$ of a real closed field $K$ is a discretely ordered subring of $K$ with minimal positive element $1$ such that, for every $x\in K$, there is $i\in I$ with $i\leq x<i+1$. Mourgues and Ressayre showed in [MR] that every real closed field has an integer part. Their construction implicitly uses the Axiom of Choice. We show that $AC$ is actually necessary to obtain the result by constructing a transitive model of $ZF$ which contains a real closed field without an integer part. Then we analyze some cases where the Axiom of Choice is not necessary for obtaining an integer part. On the way, we demonstrate that a class of questions containing the question whether the Axiom of Choice is necessary for the proof of a certain $ZFC$-theorem is algorithmically undecidable. We further apply the methods to show that it is independent of $ZF$ whether every real closed field has a value group section and a residue field section. This also sheds some light on the possibility to effectivize constructions of integer parts and value group sections which was considered e.g. in [DKKL] and [KL]

We address the study of some curvature equations for distinguished submanifolds in para-Kähler geometry. We first observe that a para-complex submanifold of a para-Kähler manifold is minimal. Next we describe the extrinsic geometry of Lagrangian submanifolds in the para-complex Euclidean space $\mathbb{D}^n$ and discuss a number of examples, such as graphs and normal bundles. We also characterize those Lagrangian surfaces of $\mathbb{D}^2$ which are minimal and have indefinite metric. Finally we describe those Lagrangian self-similar solutions of the Mean Curvature Flow (with respect to the neutral metric of $\mathbb{D}^n$) which are $SO(n)$-equivariant.

We study the non-existence of smooth curves of low degree and low positive genus on a general heptic hypersurface of $\mathbb {P}^5$. The genus $0$ case was proved for the same degrees by Hana - Johnsen and Cotterill.

We continue investigations started by Lakeland on Fuchsian and Kleinian groups which have a Dirichlet fundamental domain that also is a Ford domain in the upper half-space model of hyperbolic $2$- and $3$-space, or which have a Dirichlet domain with multiple centers. Such domains are called DF-domains and Double Dirichlet domains respectively. Making use of earlier obtained concrete formulas for the bisectors defining the Dirichlet domain of center $i \in \mathbb H^2$ or center $j \in \mathbb H^3$, we obtain a simple condition on the matrix entries of the side-pairing transformations of the fundamental domain of a Fuchsian or Kleinian group to be a DF-domain. Using the same methods, we also complement a result of Lakeland stating that a cofinite Fuchsian group has a DF domain (or a Dirichlet domain with multiple centers) if and only if it is an index $2$ subgroup of the discrete group G of reflections in a hyperbolic polygon.