
Dmitri AlekseevskyInstitute of Information Security
Dmitri Alekseevsky
Doctor Degree
About
118
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Introduction
My main areas of interest are
Differential Geometry,Lie Groups and Transformation Groups, Applications to Mathematical Physics and Neuroscience:
(Pseudo)Riemannian Geometry, Spinor Geometry, Homogeneous Geometry, Holonomy groups, Generalized Geometry, Geometric Structures, Geometry of PDE, GR and Supergravity, Geometrical Problems arising in Neuroscience and Vision (Neurogeometry).
Additional affiliations
Education
September 1958 - June 1963
Publications
Publications (118)
A class of minimal almost complex submanifolds of a Riemannian manifold \(\tilde{M}^{4n}\) with a parallel quaternionic structure Q, in particular of a 4-dimensional oriented Riemannian manifold, is studied. A notion of Kähler submanifold is defined. Any Kähler submanifold is pluriminimal. In the case of a quaternionic Kähler manifold \(\tilde{M}^{...
A homogeneous Riemannian space (M = G/H,g) is called a geodesic orbit space (shortly, GO-space) if any geodesic is an orbit of one-parameter subgroup of the isometry group G. We study the structure of compact GO-spaces and give some sufficient conditions for existence and non-existence of an invariant metric g with homogeneous geodesics on a homoge...
. On a quaternion-Kahler manifold M the Hamiltonian of a Killing field is a 2-form [GL], [S1] and we show it is an eigenform of the Laplacian corresponding to the minimal eigenvalue. This gives a quaternionic version of a famous result of Lichnerowicz and Matsushima on Kahler-Einstein geometry. As a main tool we use the twistor fibration t : Z !M a...
Let (M4n,g,Q) be a quaternion Kähler manifold with reduced scalar curvature ν = K/4n(n + 2). Suppose J is an almost complex structure which is compatible with the quaternionic structure Q and let θ = − δ F ∘ J be the Lee form of J. We prove the following local results: (1) if J is conformally symplectic, then it is parallel and ν = 0; (2) if J is c...
We propose a differential geometric model of hypercolumns in the primary visual cortex V1 that combines features of the symplectic model of the primary visual cortex by A. Sarti, G. Citti and J. Petitot and of the spherical model of hypercolumns by P. Bressloff and J. Cowan. The model is based on classical results in Conformal Geometry.
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 ma...
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...
In Communications in Contemporary Mathematics 24 3, (2022),the authors have developed a method for constructing G -invariant partial differential equations (PDEs) imposed on hypersurfaces of an $(n+1)$ -dimensional homogeneous space $G/H$ , under mild assumptions on the Lie group G . In the present paper, the method is applied to the case when $G=\...
The paper is devoted to the development of the differential geometry of saccades and saccadic cycles. We recall an interpretation of Donder’s and Listing’s law in terms of the Hopf fibration of the 3-sphere over the 2-sphere. In particular, the configuration space of the eye ball (when the head is fixed) is the 2-dimensional hemisphere \(S^+_L\), w...
A correction to this paper has been published: https://doi.org/10.1007/JHEP11(2021)100
The paper is devoted to the development of the differential geometry of saccades and saccadic cycles. We recall an interpretation of Donder's and Listing's law in terms of the Hopf fibration of the $3$-sphere over the $2$-sphere. In particular, the configuration space of the eye ball (when the head is fixed) is the 2-dimensional hemisphere $S^+_L$,...
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 \m...
By Vinberg theory any homogeneous convex cone V \mathscr {V} may be realised as the cone of positive Hermitian matrices in a T T -algebra of generalised matrices. The level hypersurfaces V q ⊂ V \mathscr {V}_{q} \subset \mathscr {V} of homogeneous cubic polynomials q q with positive definite Hessian (symmetric) form g ( q ) ≔ − Hess ( log ( q )...
We study semi-Riemannian cones admitting a parallel totally isotropic distribution of rank two. We give a local classification of the base manifolds of such holonomy.KeywordsPseudo-Riemannian manifoldsMetric conesSpecial holonomy
We study homogeneous Lorentzian manifolds M=G/L of a connected reductive Lie group G modulo a connected reductive subgroup L, under the assumption that M is (almost) G-effective and the isotropy representation is totally reducible. We show that the description of such manifolds reduces to the case of semisimple Lie groups G. Moreover, we prove that...
We study homogeneous Lorentzian manifolds $M = G/L$ of a connected reductive Lie group $G$ modulo a connected reductive subgroup $L$, under the assumption that $M$ is (almost) $G$-effective and the isotropy representation is totally reducible. We show that the description of such manifolds reduces to the case of semisimple Lie groups $G$. Moreover,...
The first part of the paper contains a short review of the image processing in early vision is static, when the eyes and the stimulus are stable, and in dynamics, when the eyes participate in fixation eye movements. In the second part, we give an interpretation of Donders’ and Listing’s law in terms of the Hopf fibration of the 3-sphere over the 2-...
We study the geometry and holonomy of semi-Riemannian, time-like metric cones that are indecomposable, i.e., which do not admit a local decomposition into a semi-Riemannian product. This includes irreducible cones, for which the holonomy can be classified, as well as non irreducible cones. The latter admit a parallel distribution of null k -planes,...
We propose a model for the hypercolumns that combines the main points of the symplectic model of the primary visual cortex V1 by A. Sarti, G. Citti and J. Petitot and of the spherical model of the hypercolumns by P. Bressloff and J. Cowan. The model by Sarti, Citti and Petitot is based on the assumption that each Gabor filter $\text{Gab}_n$, modell...
A bstract
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 o...
The article is a report on the biography and achievements of Ernest Borisovich Vinberg, an outstanding Russian mathematician, who passed away in Moscow on May 12, 2020. We discuss his contributions to various areas of mathematics such as Riemannian and Lobachevsky geometries, homogeneous convex cones, Lie groups and Invariant theory, equivariant sy...
A conformal spherical model of hypercolumns of primary visual cortex V1 is proposed. It is a modification of the Bressloff-Cowan Riemannian spherical model. The main assumption is that simple neurons of a hypercolumn, considered as Gabor filters, obtained for the mother Gabor filter by transformations from the Möbius group \(Sl(2, \mathbb {C})\). I...
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 el...
We study Lorentzian manifolds (M,g) of dimension n≥4, equipped with a maximally twisting shearfree null vector field p, for which the leaf space S=M/{exptp} is a smooth manifold. If n=2k, the quotient S=M/{exptp} is naturally equipped with a subconformal structure of contact type and, in the most interesting cases, it is a regular Sasaki manifold...
Given a Riemannian space N N of dimension n n and a field D D of symmetric endomorphisms on N N , we define the extension M M of N N by D D to be the Riemannian manifold of dimension n + 1 n+1 obtained from N N by a construction similar to extending a Lie group by a derivation of its Lie algebra. We find the conditions on N N and D D which imply th...
Let [Formula: see text] be an [Formula: see text]-dimensional homogeneous manifold and [Formula: see text] be the manifold of [Formula: see text]-jets of hypersurfaces of [Formula: see text]. The Lie group [Formula: see text] acts naturally on each [Formula: see text]. A [Formula: see text]-invariant partial differential equation of order [Formula:...
We classify the finite type (in the sense of E. Cartan theory of prolongations) subalgebras h ⊂ sp(V), where V is the symplectic 4-dimensional space, and show that they satisfy h (k) = 0 for all k > 0. Using this result, we reduce the problem of classification of graded transitive finite-dimensional Lie algebras g of symplectic vector fields on V t...
We classify contact manifolds (M,D) which are homogeneous under a connected semisimple Lie group G, and symmetric in the sense that there exists a contactomorphism of (M,D) normalizing G, fixing a point o in M and restricting to minus identity along Do.
Given a non-compact semisimple Lie group G we describe all homogeneous spaces G / L carrying an invariant almost-Kähler structure (ω,J)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \be...
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 c...
We study compact, simply connected, homogeneous 8-manifolds admitting invariant [Formula: see text]-structures, classifying all canonical presentations [Formula: see text] of such spaces, with [Formula: see text] simply connected. For each presentation, we exhibit explicit examples of invariant [Formula: see text]-structures and we describe their t...
There are several different, but equivalent definitions of geodesics in a Riemannian manifold, based on two characteristic properties: geodesics as shortest curves and geodesics as straightest curves. They are generalized to sub-Riemannian manifolds, but become non-equivalent. We give an overview of different approaches to the definition, study and...
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 $\math...
We classify contact manifolds $(M,\mathcal D)$ which are homogeneous under a connected semisimple Lie group $G$, and symmetric in the sense that there exists a contactomorphism of $(M,\mathcal D)$ normalizing $G$, fixing a point $o$ in $M$ and restricting to minus identity along $\mathcal D_o$.
Podestà and Spiro (Osaka J Math 36(4):805–833, 1999) introduced a class of G-manifolds M with a cohomogeneity one action of a compact semisimple Lie group G which admit an invariant Kähler structure (g, J) (“standard G-manifolds”) and studied invariant Kähler and Kähler–Einstein metrics on M. In the first part of this paper, we gave a combinatoric...
A Vaisman manifold is a special kind of locally conformally Kähler manifold, which is closely related to a Sasaki manifold. In this paper, we show a basic structure theorem of simply connected homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups, up to holomorphic isometry. For the case of unimodular Lie groups, we obtain a complete cl...
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 (Eu...
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...
Let $M$ be an $(n+1)$-dimensional manifold, let a group $G$ act transitively on $M$ and let $J^k(n,M)$ denote the space of $k$-jets of hypersurfaces of $M$. We make the following two assumptions on the action of $G$. First, there exists a hypersurface $\mathcal{S}_F\subset M$, referred to as a fiducial hypersurface, such that the $G$-orbit of the $...
F. Podest\`a and A. Spiro introduced a class of $G$-manifolds $M$ with a cohomogeneity one action of a compact semisimple Lie group $G$ which admit an invariant Kaehler structure $(g,J)$ (``standard $G$-manifolds") and studied invariant Kaehler and Kaehler-Einstein metrics on $M$. In the first part of this paper, we gave a combinatoric description...
Consider an oriented four-dimensional Lorentzian manifold $(\widetilde{M}^{3, 1}, \widetilde{g})$ and an oriented seven-dimensional Riemannian manifold $(M^{7}, g)$. We describe a class of decomposable eleven-dimensional supergravity backgrounds on the product manifold $({\mathcal{M}}^{10, 1}=\widetilde{M}^{3,1} \times M^7, g_{{\mathcal{M}}}=\widet...
We study compact, simply connected, homogeneous 8-manifolds admitting invariant Spin(7)-structures, classifying all canonical presentations G/H of such spaces, with G simply connected. For each presentation, we exhibit explicit examples of invariant Spin(7)-structures and we describe their type, according to Fern\'andez classification. Finally, we...
We study the geometry and holonomy of semi-Riemannian, time-like metric cones that are indecomposable, i.e., which do not admit a local decomposition into a semi-Riemannian product. This includes irreducible cones, for which the holonomy can be classified, as well as non irreducible cones. The latter admit a parallel distribution of null $k$-planes...
Given a non compact semisimple Lie group $G$ we describe all homogeneous spaces $G/L$ carrying an invariant almost K\"ahler structure $(\omega,J)$. When $L$ is abelian and $G$ is of classical type, we classify all such spaces which are Chern-Einstein, i.e. which satisfy $\rho = \lambda\omega$ for some $\lambda\in\mathbb R$, where $\rho$ is the Ricc...
Given a Riemannian space $N$ of dimension $n$ and a field $D$ of symmetric endomorphisms on $N$, we define the extension $M$ of $N$ by $D$ to be the Riemannian manifold of dimension $n+1$ obtained from $N$ by a construction similar to extending a Lie group by a derivation of its Lie algebra. We find the conditions on $N$ and $D$ which imply that th...
In this paper we develop some basic strategies to classify homogeneous locally conformally Kaehler and Sasaki manifolds. In particular, we make a complete classification of simply connected homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups, up to isomorphisms.
We study finite-dimensional transitive Lie algebras $\mathfrak g$ of symplectic vector fields on the symplectic 4-dimensional space $V$. We classify the linear isotropy algebras $\mathfrak h \subset \mathfrak{sp}(V)$ which are of finite type and, under some assumptions, the transitive Lie algebras $\mathfrak g$ of symplectic vector fields on $V$ wh...
Consider an oriented four-dimensional Lorentzian manifold $(\widetilde{M}^{3, 1}, \widetilde{g})$ and an oriented seven-dimensional Riemannian manifold $(M^{7}, g)$. We describe a class of decomposable eleven-dimensional supergravity backgrounds on the product manifold $({\mathcal{M}}^{10, 1}=\widetilde{M}^{3,1} \times M^7, g_{{\mathcal{M}}}=\widet...
We study higher dimensional versions of shearfree null-congurences in conformal Lorentzian manifolds. We show that such structures induce a subconformal structure and a partially integrable almost CR-structure on the leaf space and we classify the Lorentzian metrics that induce the same subconformal structure. In the last section we survey some kno...
Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description of the structure of these curv...
Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description of the structure of these curv...
For each simple Lie algebra (Formula presented.) (excluding, for trivial reasons, type (Formula presented.)), we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in (Formula presented.), a homogeneous contact manifold. Here a PDE (Formula presented.) has degree (Formula presented.) if (Formula presented.) is...
Let M be a cohomogeneity one manifold of a compact semisimple Lie group G with one singular orbit \(S_0 = G/H\). Then M is G-diffeomorphic to the total space \(G \times _H V\) of the homogeneous vector bundle over \(S_0\) defined by a sphere transitive representation of G in a vector space V. We describe all such manifolds M which admit an invarian...
Let $M$ be a cohomogeneity one manifold of a compact semisimple Lie group $G$ with one singular orbit $S_0 = G/H$. Then $M$ is $G$- diffeomorphic to the total space $G \times_H V$ of the homogeneous vector bundle over $S_0$ defined by a sphere transitive representation of $G$ in a vector space $V$. We describe all such manifolds $M$ which admit an...
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 manifol...
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 mi...
The classical theory of prolongations 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 structur...
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 structure...
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 spi...
A depth one grading $\mathfrak{g}= \mathfrak{g}^{-1}\oplus \mathfrak{g}^0
\oplus \mathfrak{g}^1 \oplus \cdots \oplus \mathfrak{g}^{\ell}$ of a finite
dimensional Lie superalgebra $\mathfrak{g}$ is called nonlinear irreducible if
the isotropy representation $\mathrm{ad}_{\mathfrak{g}^0}|_{\mathfrak{g}^{-1}}$
is irreducible and $\mathfrak{g}^1 \neq (...
Let \gh = \gh_{-k}\oplus \cdots \oplus \gh_{l} (k >0, l \geq 0) be a finite
dimensional graded Lie algebra, with an Euclidian Ad_{G}-invariant (Lie(G) =
\gh_{0}) 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...
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-invari...
Let (M,g,Q) be a simply connected, complete, quaternionic Kähler manifold without flat de Rham factor. Then any 1-parameter group of transformations of M which preserve the quaternionic structure Q preserves also the metric g. Moreover, if (M,g) is irreducible then the quaternionic Kähler metric g on (M,Q) is unique up to a homothety.
A complete classification of quaternionic Riemannian spaces (that is, spaces with the holonomy group , ) which admit a transitive solvable group of motions is given. It turns out that the rank of these spaces does not exceed four and that all spaces whose rank is less than four are symmetric. The spaces of rank four are in natural one-to-one corres...
A pseudo-Riemannian symmetric space M = G/H of dimension 4n > 4 is called quaternionic Kähler if its holonomy group Hol is a subgroup of Sp 1 · Sp p,q , where p + q = n, and it is called para-quaternionic Kähler if Hol ⊂ Sp 1 (R) · Sp n (R). We study the structure of pseudo-Riemannian quaternionic Kähler and para-quaternionic Kähler symmetric space...
We classify all compact simply connected homogeneous CR manifolds M of codimension one and with non-degenerate Levi form up
to CR equivalence. The classification is based on our previous results and on a description of the maximal connected compact
group G(M) of automorphisms of M. We characterize also the standard homogeneous CR manifolds as the h...
A geodesic of a homogeneous Riemannian manifold (M=G/K,g) is called homogeneous if it is an orbit of an one-parameter subgroup of G. In the case when M=G/H is a naturally reductive space, that is the G-invariant metric g is defined by some non degenerate biinvariant symmetric bilinear form B, all geodesics of M are homogeneous. We consider the case...
If G is a transitive Lie group of quasi-isometries of a Riemannian manifold (M,g), the growth-type of G is equal to the Riemannian growth of (M,g).
On an almost quaternionic manifold (M4n;Q ) we study the inte- grability of almost complex structures which are compatible with the almost quaternionic structure Q.I f n2, we prove that the existence of two com- patible complex structures I1;I26 =I1 forces (M4n;Q) to be quaternionic. If n =1 , that is ( M 4;Q )=( M 4 ; ( g );or) is an oriented conf...
Let (M, g, Q) be a simply connected, complete, quaternionic Kähler manifold without flat de Rham factor. Then any 1-parameter group of transformations of M which preserve the quaternionic structure Q preserves also the metric g. Moreover, if (M, g) is irreducible then the quaternionic Kähler metric g on (M, Q) is unique up to a homothety.
Let (M,Q) be a quaternionic manifold. Conditions for existence of hypercomplex structures H subordinated to the quaternionic structure Q are determined, in particular for a quaternionic Kähler manifold (M,g,Q). Some special systems of almost hypercomplex structures which are admissible for Q are also considered and their relationships with quaterni...
This is a report on some recent results concerning six quaternionic-like structures that are defined on a manifold M (almost quaternionic, hypercomplex, unimodular quaternionic, unimodular hypercomplex, Hermitian quaternionic, Hermitian hypercomplex). The interrelations between them and their automorphism groups are considered in the framework of t...
. We consider quaternionic transformations of a quaternionic Kahler manifold (M; g; Q). General conditions for unicity of the quaternionic Kahler structure (g; Q) for given Q are applied to the case of two quaternionic Kahler metrics which are in correspondence through a quaternionic transformation. Characterization of compact quaternionic Kahler m...