Papers by Дмитрий (Dmitri) Алексеевский (Alekseevsky)
Recent Developments in Pseudo-Riemannian Geometry
Riemannian and pseudo-Riemannian symmetric spaces with semisimple transvection group are known an... more Riemannian and pseudo-Riemannian symmetric spaces with semisimple transvection group are known and classified for a long time. Contrary to that the description of pseudo-Riemannian symmetric spaces with non-semisimple transvection group is an open problem. In the last years some progress on this problem was achieved. In this article we want to explain these results and some of their applications. Contents * Supported by Heisenberg program, DFG.
arXiv (Cornell University), Jun 26, 2023
The paper is devoted to the development of the differential geometry of saccades and saccadic cyc... more 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 , which is called Listing's hemisphere. We give three characterizations of saccades: as geodesic segment ab in the Listing's hemisphere, as the gaze curve and as a piecewise geodesic curve of the orthogonal group. We study the geometry of saccadic cycle, which is represented by a geodesic polygon in the Listing hemisphere, and give necessary and sufficient conditions, when a system of lines through the center of eye ball is the system of axes of rotation for saccades of the saccadic cycle, described in terms of world coordinates and retinotopic coordinates. This gives an approach to the study the visual stability problem.
Lecture Notes in Computer Science, 2023
Annals of Global Analysis and Geometry
Cohomogeneity one Pdemannian G-manifolds (i.e. [,iemannian manifolds with a group G of isometrics... more Cohomogeneity one Pdemannian G-manifolds (i.e. [,iemannian manifolds with a group G of isometrics having an orbit of eodimension one) are s~udied. A description of such manifolds (up to some normal equivalence) is given in terms of Lie subgroups of Lie group G. The twist of a geodesic normal to all orbits is defined as the n~mber of intersections with a singular orbit. It is equal to the order of some Weyl group, associated with the G-manifold. Some results about possible values of the twist are obtained.
Journal of High Energy Physics
A correction to this paper has been published: https://doi.org/10.1007/JHEP11(2021)100
Springer proceedings in mathematics & statistics, 2022
arXiv (Cornell University), Jan 3, 2023
By Vinberg theory any homogeneous convex cone V may be realised as the cone of positive Hermitian... more By Vinberg theory any homogeneous convex cone V may be realised as the cone of positive Hermitian matrices in a T-algebra of generalised matrices. The level hypersurfaces Vq ⊂ V of homogeneous cubic polynomials q with positive definite Hessian (symmetric) form g(q) := − Hess(log(q))| T Vq are the 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 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 admissible (i.e. such that the hypersurface Vq = {q = 1} ∩ 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.
Algebra and Analysis, 1996
Let (g, J) be an invariant Hermitian structure on a full flag manifold. We prove that if the Kahl... more Let (g, J) be an invariant Hermitian structure on a full flag manifold. We prove that if the Kahler form \omega satisfies d (d\omega \circ J^{3}) =0, then (g, J) is Kahler. We apply this result to generalized Kahler geometry.
American Mathematical Society Translations: Series 2, 2003
Journal of Geometry and Physics, 1990
The hyper-Kahler Calabi-Yau metric m on the Fermat surface associated with the embedding F C ¢. p... more The hyper-Kahler Calabi-Yau metric m on the Fermat surface associated with the embedding F C ¢. pS is studied. We prove that the lattice of integer parallel 2-forms on the Riemannian manifold (F, m) has the Gram matn'x~ diag(4,8,8). We use itfor calculation of the isometry group Isom(m). The action of this group on the twistor space of parallel complex structure on (F, ra) is described and the existence of 10 complex structures with non-trivial stabilizer in Isom (m) is established. Then we give the classifcation of all connected 2 dimensional totally geodesic submanifolds which are fixed points sets of isometries. There are 288 such manifolds of genus 0,1,2,3,5. They are complex curves respect to one of the 5 (up to a sign) distinguished complex structures. From the physical point of view such submanifolds are interpreted as (holomorphic) instantons for sigma model with the value in K3 surface. Such instantons are studied by physicistis in relation with string theory.(*) The generalization of the results to some class of Calabi-Yan metn'cs on K3 surfaces X, associated with the embeddings X C UP is given.
Journal of Geometry and Physics, 1993
The notion ofa group oftwistor type is defined as a linear group G whose Lie algebra has an eleme... more The notion ofa group oftwistor type is defined as a linear group G whose Lie algebra has an element J with J=-1. The twistor space Z of a G-structure with a connection on a manifold M is constructed where G is a Lie group of twistor type. It is the total space ofa bundle over M with a complex affine symmetric space G/H as a fiber. A natural almost complex structure J and a horizontal distribution Hon Z are defined and studied. The conditions of integrability ofJ and holomorphicity of H reduce to some linear conditions on the curvature tensor of the connection. They may be considered as generalized self-dual equations. It is shown that for some Lie group G these equations are fulfilled automatically. For other groups G there are some obstacles that are described in termsof a decomposition ofthe curvature tensor associated with a bigradation of the appropriate Koszul-Spencer complex.
Итоги науки и техники. Серия, Современные проблемы математики, фундаментальные направления, 2022
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Papers by Дмитрий (Dmitri) Алексеевский (Alekseevsky)