arXiv subjects

Dmitri V. Alekseevsky

Publications and source records attributed to Dmitri V. Alekseevsky.

At least 19 recordsLinked to original sources

arXiv2025

Conformally homogeneous Lorentzian spaces

We prove that if a 1-connected non-conformally flat conformal Lorentzian manifold $(M,c)$ admits a connected essential transitive group of conformal transformations, then there exists a metric $g\in c$ such that $(M,g)$ is a complete homogeneous plane wave. This finishes the classification of 1-connected Lorentzian manifolds, which admit transitive essential conformal group. We also prove that the group of conformal transformations of a non-conformally flat 1-connected homogeneous plane wave $(M,g)$ consists of homotheties, and it is a 1-dimensional extension of the group of isometries.

math.DG↗

arXiv2024

The functional architecture of the early vision and neurogeometric models

The initial sections of the paper give a concise presentation, specially designed for a mathematically oriented audience, of some of the most basic facts on the functional architecture of early vision. Such information is usually scattered in a variety of papers and books, which are not easily accessible by non-specialists. Our goal is thus to offer a handy and short introduction to this topics, which might be helpful for researchers willing to enter the area of the applications of modern Differential Geometry in studies on the visual systems, baptized neurogeometry by J. Petitot. We then offer a survey of three of the most important neurogeometric models: Petitot's contact model of the primary visual cortex, its extension to A. Sarti, G. Citti and J. Petitot's symplectic model, and P. C. Bressloff and J. D. Cowan's spherical model of hypercolumns. We finally discuss the main points of the so-called ``conformal model'' for hypercolumns (a model that was briefly presented in [D. V. Alekseevsky, Conformal model of hypercolumns in V1 cortex and the Moebius group, in ``Geometric science of information'', pp. 65--72, Springer, 2021] and given in detail in [D. V. Alekseevsky and A. Spiro, Conformal models for hypercolumns in the primary visual cortex V1, arXiv 2024]), which can be considered as a synthesis of the symplectic and the spherical models.

q-bio.NC↗

arXiv2024

Projectively and affinely invariant PDEs on hypersurfaces

In [Alekseevsky, Gutt, Manno, Moreno: "A general method to construct invariant PDEs on homogeneous manifolds", Communications in Contemporary Mathematics (2021)] the authors have developed a method for constructing $G$-invariant PDEs imposed on hypersurfaces of an $(n+1)$-dimensional homogeneous space $G/H$, under mild assumptions on the Lie groups $G$. In the present paper the method is applied to the case when $G=\mathsf{PGL}(n+1)$ or $G=\mathsf{Aff}(n+1)$ and the homogeneous space $G/H$ is the $(n+1)$-dimensional projective $\mathbb{P}^{n+1}$ or affine $\mathbb{A}^{n+1}$ space, respectively. The paper's main result is that projectively or affinely invariant PDEs with $n$ independent and one unknown variables are in one-to-one correspondence with $\mathsf{CO}(d,n-d)$-invariant hypersurfaces of the space of trace-free cubic forms in $n$ variables. Local descriptions are also provided.

math.DG↗

arXiv2023

The Levi-Civita connections of manifolds with prescribed optical geometries

We explicitly derive the Christoffel symbols in terms of adapted frame fields for the Levi-Civita connection of a Lorentzian $n$-manifold $(M, g)$, equipped with a prescribed optical geometry of Kähler-Sasaki type. The formulas found in this paper have several important applications, such as determining the geometric invariants of Lorentzian manifolds with prescribed optical geometries or solving curvature constraints.

math.DG↗

arXiv2023

Special Vinberg Cones and the Entropy of BPS Extremal Black Holes

We consider the static, spherically symmetric and asymptotically flat BPS extremal black holes in ungauged N = 2 D = 4 supergravity theories, in which the scalar manifold of the vector multiplets is homogeneous. By a result of Shmakova on the BPS attractor equations, the entropy of this kind of black holes can be expressed only in terms of their electric and magnetic charges, provided that the inverse of a certain quadratic map (uniquely determined by the prepotential of the theory) is given. This inverse was previously known just for the cases in which the scalar manifold of the theory is a homogeneous symmetric space. In this paper we use Vinberg's theory of homogeneous cones to determine an explicit expression for such an inverse, under the assumption that the scalar manifold is homogeneous, but not necessarily symmetric. As immediate consequence, we get a formula for the entropy of BPS black holes that holds in any model of N = 2 supergravity with homogeneous scalar manifold.

hep-th↗

arXiv2023

Special Vinberg cones, invariant admissible cubics and special real manifolds

By Vinberg theory any homogeneous convex cone $\mathcal V$ may be realized as the cone of positive Hermitian matrices in a $T$-algebra of generalised matrices. The level hypersurfaces $\mathcal V_{q} \subset \mathcal V$ of homogeneous cubic polynomials $q$ with positive definite Hessian (symmetric) form $g_q := - \operatorname{Hess}(\log(q))|_{T \mathcal V_q}$ are the {\it special real manifolds}. Such manifolds occur as scalar manifolds of the vector multiplets in $N=2$, $D=5$ supergravity and, through the $r$-map, correspond to Kähler scalar manifolds in $N = 2$ $D = 4$ supergravity. We offer a simplified exposition of the Vinberg theory in terms of $\operatorname{Nil}$-algebras (= the subalgebras of upper triangular matrices in Vinberg $T$-algebras) and we use it to describe all rational functions on a special Vinberg cone that are $G_0$- or $G'$- invariant, where $G_0$ is the unimodular subgroup of the solvable group $G$ acting simply transitively on the cone, and $G'$ is the unipotent radical of $G_0$. The results are used to determine $G_0$- and $G'$-invariant cubic polynomials $q$ that are {\it admissible} (i.e. such that the hypersurface $ \mathcal V_q=\{ q=1\}\cap \mathcal V $ has positive definite Hessian form $g_q$) for rank $2$ and rank $3$ special Vinberg cones. We get in this way examples of continuous families of non-homogeneous special real manifolds of cohomogeneity less than or equal to two.

math-ph↗

arXiv2021

Lorentzian manifolds with shearfree congruences and Kähler-Sasaki geometry

We study Lorentzian manifolds $(M, g)$ of dimension $n\geq 4$, equipped with a maximally twisting shearfree null vector field $p_o$, for which the leaf space $S = M/\{\exp t p_o\}$ is a smooth manifold. If $n = 2k$, the quotient $S = M/\{\exp t p_o\}$ is naturally equipped with a subconformal structure of contact type and, in the most interesting cases, it is a regular Sasaki manifold projecting onto a quantisable Kähler manifold of real dimension $2k -2$. Going backwards through this line of ideas, for any quantisable Kähler manifold with associated Sasaki manifold $S$, we give the local description of all Lorentzian metrics $g$ on the total spaces $M$ of $A$-bundles $π: M \to S$, $A = S^1, \mathbb R$, such that the generator of the group action is a maximally twisting shearfree $g$-null vector field $p_o$. We also prove that on any such Lorentzian manifold $(M, g)$ there exists a non-trivial generalized electromagnetic plane wave having $p_o$ as propagating direction field, a result that can be considered as a generalization of the classical $4$-dimensional Robinson Theorem. We finally construct a 2-parametric family of Einstein metrics on a trivial bundle $M = \mathbb R \times S$ for any prescribed value of the Einstein constant. If $\dim M = 4$, the Ricci flat metrics obtained in this way are the well-known Taub-NUT metrics.

math.DG↗

arXiv2020

A general method to construct invariant PDEs on homogeneous manifolds

Let $M = G/H$ be an $(n+1)$-dimensional homogeneous manifold and $J^k(n,M)=:J^k$ be the manifold of $k$-jets of hypersurfaces of $M$. The Lie group $G$ acts naturally on each $J^k$. A $G$-invariant PDE of order $k$ for hypersurfaces of $M$ (i.e., with $n$ independent variables and $1$ dependent one) is defined as a $G$-invariant hypersurface $\mathcal{E} \subset J^k$. We describe a general method for constructing such invariant PDEs for $k\geq 2$. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup $H^{(k-1)}$ of the $(k-1)$-prolonged action of $G$. We apply this approach to describe invariant PDEs for hypersurfaces in the Euclidean space $\mathbb{E}^{n+1 }$ and in the conformal space $\mathbb{S}^{n+1}$. Our method works under some mild assumptions on the action of $G$, namely: A1) the group $G$ must have an open orbit in $J^{k-1}$, and A2) the stabilizer $H^{(k-1)}\subset G$ of the fibre $J^k\to J^{k-1}$ must factorize via the group of translations of the fibre itself.

math.DG↗

arXiv2020

Shortest and Straightest Geodesics in Sub-Riemannian Geometry

There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as (locally) shortest curves (Euler-Lagrange, Pontyagin and Hamilton) and three definitions of geodesics as straightest curves (d'Alembert , Levi-Civita-Schouten and Cartan-Tanaka ), used in nonholonomic mechanics and discuss their interrelations. We consider a big class of sub-Riemannian manifolds associated with principal bundle over a Riemannian manifolds, for which shortest geodesics coincides with straightest geodesics. Using the geometry of flag manifolds, we describe some classes of compact homogeneous sub-Riemannian manifolds (including contact sub-Riemannian manifolds and symmetric sub-Riemannian manifolds) where straightest geodesics coincides with shortest geodesics. Construction of geodesics in these cases reduces to description of Riemannian geodesics of the Riemannian homogeneous manifold or left-invariant metric on a Lie group.

math.DG↗

arXiv2018

Homogeneous almost Kähler manifolds and the Chern-Einstein equation

Given a non compact semisimple Lie group $G$ we describe all homogeneous spaces $G/L$ carrying an invariant almost Kähler structure $(ω,J)$. When $L$ is abelian and $G$ is of classical type, we classify all such spaces which are Chern-Einstein, i.e. which satisfy $ρ= λω$ for some $λ\in\mathbb R$, where $ρ$ is the Ricci form associated to the Chern connection.

math.DG↗

arXiv2017

Lowest degree invariant 2nd order PDEs over rational homogeneous contact manifolds

For each simple Lie algebra $\mathfrak{g}$ (excluding, for trivial reasons, type ${\sf C}$) we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in $\mathbb{P}\mathfrak{g}$, a homogeneous contact manifold. Here a PDE $F(x^i,u,u_i,u_{ij})=0$ has degree $\le d$ if $F$ is a polynomial of degree $\le d$ in the minors of $(u_{ij})$, with coefficients functions of the contact coordinates $x^i$, $u$, $u_i$ (e.g., Monge-Ampère equations have degree 1). For $\mathfrak{g}$ of type ${\sf A}$ or ${\sf G}$ we show that this gives all invariant second-order PDEs. For $\mathfrak{g}$ of type ${\sf B}$ and ${\sf D}$ we provide an explicit formula for the lowest-degree invariant second-order PDEs. For $\mathfrak{g}$ of type ${\sf E}$ and ${\sf F}$ we prove uniqueness of the lowest-degree invariant second-order PDE; we also conjecture that uniqueness holds in type ${\sf D}$.

math.DG↗

arXiv2016

Spin structures on compact homogeneous pseudo-Riemannian manifolds

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic structure. Then we investigate spin structures on principal torus bundles over flag manifolds, i.e. C-spaces, or equivalently simply-connected homogeneous complex manifolds M=G/L of a compact semisimple Lie group G. We study the topology of M and we provide a sufficient and necessary condition for the existence of an (invariant) spin structure, in terms of the Koszul form of F. We also classify all C-spaces which are fibered over an exceptional spin flag manifold and hence they are spin.

math.DG↗

arXiv2016

Lorentzian manifolds with transitive conformal group

We study pseudo-Riemanniasn manifolds $(M,g)$ with transitive group of conformal transformation which is essential, i.e. does not preserves any metric conformal to $g$. All such manifolds of Lorentz signature with non exact isotropy representation of the stability subalgebra are described. A construction of essential conformally homogeneous manifolds with exact isotropy represenatation is given. Using spinor formalism, we prove that it provides all 4-dimensional non conformally flat Lorentzian 4-dimensional manifolds with transitive essentially conformal group.

math.DG↗

arXiv2016

Prolongation of Tanaka structures: an alternative approach

The classical theory of prolongation of G-structures was generalized by N. Tanaka to a wide class of geometric structures (Tanaka structures), which are defined on a non-holonomic distribution. Examples of Tanaka structures include subriemannian, subconformal, CR-structures, structures associated to second order differential equations and structures defined by gradings of Lie algebras (in the setting of parabolic geometry). Tanaka's prolongation procedure associates to a Tanaka structure of finite order a manifold with an absolute parallelism. It is a very fruitful method for the description of local invariants, investigation of the automorphism group and the equivalence problem. In this paper we develop an alternative constructive approach for Tanaka's prolongation procedure, based on the theory of quasi-gradations in filtered vector spaces, G-structures and their torsion functions.

math.DG↗

arXiv2015

Quaternionic Kähler metrics associated with special Kähler manifolds

We give an explicit formula for the quaternionic Kähler metrics obtained by the HK/QK correspondence. As an application, we give a new proof of the fact that the Ferrara-Sabharwal metric as well as its one-loop deformation is quaternionic Kähler. A similar explicit formula is given for the analogous (K/K) correspondence between Kähler manifolds endowed with a Hamiltonian Killing vector field. As an example, we apply this formula in the case of an arbitrary conical Kähler manifold.

math.DG↗

arXiv2015

Homogeneous locally conformally Kaehler and Sasaki manifolds

We prove various classification results for homogeneous locally conformally symplectic manifolds. In particular, we show that a homogeneous locally conformally Kaehler manifold of a reductive group is of Vaisman type, if the normalizer of the isotropy group is compact. We also show that such a result does not hold in the case of noncompact normalizer and determine all left-invariant locally conformally Kaehler structures on reductive Lie groups.

math.DG↗

arXiv2015

Tanaka structures (non holonomic G-structures) and Cartan connections

Let \gh = \gh_{-k}\oplus \cdots \oplus \gh_{l} (k >0, l \geq 0) be a finite dimensional real graded Lie algebra, with a Euclidian metric \langle \cdot , \cdot \rangle adapted to the gradation. The metric \langle\cdot , \cdot \rangle is called admissible if the codifferentials \partial^{*} : C^{k+1}(\gh_{-}, \gh ) \ra C^{k} (\gh_{-}, \gh) (k\geq 0) are Ad_{Q}-invariant (Lie(Q) = \gh_{0}\oplus \gh_{+}). We find necessary and sufficient conditions for a Euclidian metric, adapted to the gradation, to be admissible, and we develop a theory of normal Cartan connections, when these conditions are satisfied. We show how the treatment by A. Cap and J. Slovak (Parabolic Geometry I, Mathematical Surveys and Monographs, vol. 154, 2009), about normal Cartan connections of semisimple type, fits into our theory. We also consider in some detail the case when \gh = t^{*} (\gg ) is the cotangent Lie algebra of a non-positively graded Lie algebra \gg.

math.DG↗

arXiv2014

Homogeneous almost complex structures in dimension 6 with semi-simple isotropy

We classify invariant almost complex structures on homogeneous manifolds of dimension 6 with semi-simple isotropy. Those with non-degenerate Nijenhuis tensor have the automorphism group of dimension either 14 or 9. An invariant almost complex structure with semi-simple isotropy is necessarily either of specified 6 homogeneous types or a left-invariant structure on a Lie group. For integrable invariant almost complex structures we classify all compatible invariant Hermitian structures on these homogeneous manifolds, indicate their integrability properties (Kahler, SNK, SKT) and mark the other interesting geometric properties (including the Gray-Hervella type).

math.DG↗