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Let $(M,g)$ be a 4-dimensional Einstein Riemannian manifold. At each point $p$ of $M$, the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor $R$ at $p$. In this basis, up to standard symmetries and antisymmetries, just $5$ components of the curvature tensor $R$ are nonzero. For the space of constant curvature, the group $\mathrm{O}\left(4\right)$ acts as a transformation group between ST bases at $T_{p}M$ and for the so-called 2-stein curvature tensors, the group $\mathrm{Sp}\left(1\right)\subset \mathrm{SO}\left(4\right)$ acts as a transformation group...

The concept of geodesic graph is generalized from Riemannian geometry to Finsler geometry, in particular to homogeneous Randers g.o. manifolds. On modified H-type groups which admit a Riemannian g.o. metric, invariant Randers g.o. metrics are determined and geodesic graphs in these Finsler g.o. manifolds are constructed. New structures of geodesic graphs are observed.

In the recent paper [Yan, Z.: Existence of homogeneous geodesics on homogeneous Finsler spaces of odd dimension, Monatsh. Math. 182,1, 165–171 (2017)], it was claimed that any homogeneous Finsler space of odd dimension admits a homogeneous geodesic through any point. However, the proof contains a serious gap. The situation is a bit delicate, because the statement is correct. In the present paper, the incorrect part in this proof is indicated. Further, it is shown that homogeneous geodesics in homogeneous...

The existence of a homogeneous geodesic in homogeneous Finsler manifolds was investigated and positively answered in previous papers. It is conjectured that this result can be improved, namely that any homogeneous Finsler manifold admits at least two homogenous geodesics. Examples of homogeneous Randers manifolds admitting just two homogeneous geodesics are presented.

Structure of geodesic graphs in special families of invariant weakly symmetric Finsler metrics on modified H-type groups is investigated. Geodesic graphs on modified H-type groups with the center of dimension $1$ or $2$ are constructed. The new patterns of algebraic complexity of geodesic graphs are observed.

A broad family of involutive birational transformations of an open dense subset of ${ℝ}^n$ onto itself is constructed explicitly. Examples with arbitrarily high complexity are presented. Construction of birational transformations such that ${\phi }^k=\mathrm{Id}$ for a fixed integer $k>2$ is also presented.

The question how many real analytic affine connections exist locally on a smooth manifold $M$ of dimension $n$ is studied. The families of general affine connections with torsion and with skew-symmetric Ricci tensor, or symmetric Ricci tensor, respectively, are described in terms of the number of arbitrary functions of $n$ variables.

The question how many real analytic equiaffine connections with arbitrary torsion exist locally on a smooth manifold $M$ of dimension $n$ is studied. The families of general equiaffine connections and with skew-symmetric Ricci tensor, or with symmetric Ricci tensor, respectively, are described in terms of the number of arbitrary functions of $n$ variables.

In ( Dušek, Z., Kowalski, O. and Nikčević, S. Ž., New examples of Riemannian g.o. manifolds in dimension 7, Differential Geom. Appl. 21 (2004), 65–78.), the present authors and S. Nikčević constructed the 2-parameter family of invariant Riemannian metrics on the homogeneous manifolds $M=\left[\mathrm{SO}\right(5)\times \mathrm{SO}(2\left)\right]/\mathrm{U}\left(2\right)$ and $M=\left[\mathrm{SO}\right(4,1)\times \mathrm{SO}(2\left)\right]/\mathrm{U}\left(2\right)$. They proved that, for the open dense subset of this family, the corresponding Riemannian manifolds are g.o. manifolds which are not naturally reductive. Now we are going to investigate the remaining metrics...

For studying homogeneous geodesics in Riemannian and pseudo-Riemannian geometry (on reductive homogeneous spaces) there is a simple algebraic formula which works, at least potentially, in every given case. In the affine differential geometry, there is not such a universal formula. In the previous work, we proposed a simple method of investigation of homogeneous geodesics in homogeneous affine manifolds in dimension 2. In the present paper, we use this method on certain classes of homogeneous connections...

A homogeneous Riemannian manifold $M=G/H$ is called a “g.o. space” if every geodesic on $M$ arises as an orbit of a one-parameter subgroup of $G$. Let $M=G/H$ be such a “g.o. space”, and $m$ an $\text{Ad}\left(H\right)$-invariant vector subspace of $\text{Lie}\left(G\right)$ such that $\text{Lie}\left(G\right)=m\oplus \text{Lie}\left(H\right)$. A is a map $\xi :m\to \text{Lie}\left(H\right)$ such that $$t\mapsto exp\left(t\right(X+\xi \left(X\right)\left)\right)\left(eH\right)$$ is a geodesic for every $X\in m\setminus \left\{0\right\}$. The author calculates explicitly such geodesic graphs for certain special 2-step nilpotent Lie groups. More precisely, he deals with “generalized Heisenberg groups” (also known as “H-type groups”) whose center has dimension not...