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We prove that for an embedded unstable one-sided minimal hypersurface of the $(n+1)$-dimensional real projective space, the Morse index is at least $n+2$, and this bound is attained by the cubic isoparametric minimal hypersurfaces. We also…

We completely determine, up to homeomorphism, which simply connected compact oriented 4-manifolds admit scalar-flat, anti-self-dual Riemannian metrics. The key new ingredient is a proof that the connected sum of five reverse-oriented…

We prove existence and regularity results for the problem of maximization of one Laplace eigenvalue with respect to metrics of same volume lying in a conformal class of a Riemannian manifold of dimension $n\geq 3$.

We prove that for each dimension not less than five there exists a contraction between solvable Lie algebras that can be realized only with matrices whose Euclidean norms necessarily approach infinity at the limit value of contraction…

Fino and Kath determined all possible holonomy groups of seven-dimensional pseu\-do-Rie\-man\-nian manifolds contained in the exceptional, non-compact, simple Lie group $\mathrm{G}_2^*$ via the corresponding Lie algebras. They are…

We give the complete classification of left-invariant sub-Riemannian structures on three dimensional Lie groups in terms of the basic differential invariants. This classifications recovers other known classification results in the…

Let (N,g) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their `almost' versions). We define a left invariant Riemannian metric on N compatible with g to be minimal,…

We consider and resolve the gap problem for almost quaternion-Hermitian structures, i.e. we determine the maximal and submaximal symmetry dimensions, both for Lie algebras and Lie groups, in the class of almost quaternion-Hermitian…

A projective structure is Weyl metrizable if it has a representative that preserves a conformal structure. We interpret Weyl metrizability of 3-dimensional projective structures as certain 5-dimensional nondegenerate CR submanifolds in a…

Let $\mathbb{F}$ be a field, and $n \geq p \geq r>0$ be integers. In a recent article, Rubei has determined, when $\mathbb{F}$ is the field of real numbers, the greatest possible dimension for an affine subspace of $n$--by--$p$ matrices…

In this paper, we focus on $(n+3)$-dimensional metric $n$-Lie algebras. To begin with, we give some properties on $(n+3)$-dimensional $n$-Lie algebras. Then based on the properties, we obtain the classification of $(n+3)$-dimensional metric…

We formalize the notion of limit of an inverse system of metric spaces with $1$-Lipschitz projections having unbounded fibers. The purpose is to use sub-Riemannian groups for metrizing the space of signatures of rectifiable paths in…

The Schur $\mathsf{Lie}$-multiplier of Leibniz algebras is the Schur multiplier of Leibniz algebras defined relative to the Liezation functor. In this paper, we study upper bounds for the dimension of the Schur $\mathsf{Lie}$-multiplier of…

A basic question in submanifold theory is whether a given isometric immersion $f\colon M^n\to\R^{n+p}$ of a Riemannian manifold of dimension $n\geq 3$ into Euclidean space with low codimension $p$ admits, locally or globally, a genuine…

Let $\Bbbk$ be a field of characteristic zero and let $\mathbb{W}_n = \operatorname{Der}(\Bbbk[x_1,\cdots,x_n])$ be the $n^{\text{th}}$ general Cartan type Lie algebra. In this paper, we study Lie subalgebras $L$ of $\mathbb{W}_n$ of…

There are studied Lie groups considered as almost hypercomplex Hermitian-Norden manifolds, which are integrable and have the lowest dimension four. It is established a correspondence of the derived Lie algebras of types of invariant…

Degree of mobility of a (pseudo-Riemannian) metric is the dimension of the space of metrics geodesically equivalent to it. We describe all possible values of the degree of mobility on a simply connected n-dimensional manifold of lorentz…

For any smooth Riemannian metric on an $(n+1)$-dimensional compact manifold with boundary $(M,\partial M)$ where $3\leq (n+1)\leq 7$, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by…

In this paper we extend to non-compact Riemannian manifolds with boundary the use of two important tools in the geometric analysis of compact spaces, namely, the weak maximum principle for subharmonic functions and the integration by parts.…

Recently, the first author together with Jens Marklof studied generalizations of the classical three distance theorem to higher dimensional toral rotations, giving upper bounds in all dimensions for the corresponding numbers of distances…