2026 №57 (1)

This paper is dedicated to the outstanding geometer Leonid Evgenievich Evtushik, who was a member of the editorial board of the journal DGMF from 1990 to 2013.

The paper presents L. E. Evtushik’s biography and the main stages of his scientific work. His research interests included prob­lems of modern geometry, connection theory, the theory of diffe­ren­tial equations, and higher-order Lagrangians.

L. E. Evtushik taught the following courses at Moscow State University: “Mathematical Analysis”, “Linear Algebra and Analyt­ic Geometry”, “Geometric Structures of Analysis on Manifolds”, “Invariant Methods in Geometry and Analysis”, “Lagrangians and Higher-Order Hamiltonian Systems”, “Connection Theory”, and “Hig­her-Order Structures”.

L. E. Evtushik supervised 14 PhD students and authored 66 scien­tific papers.

In this short note, we study six-dimensional planar submani­folds of Cayley algebra on which the so-called Brown — Gray three-fold vector cross products in the octave algebra induce a Her­mitian structure. Such submanifolds of the octave algebra were int­roduced into consideration by the Russian geometer V. F. Kiri­chenko in the 90s of the last century. In such six-dimensional pla­nar Hermitian submanifolds, we consider geodesic points, that is, points at which the configuration tensor of the submanifold va­nishes. We establish that at geodesic points of six-dimensional pla­nar Hermitian submanifolds of the octave algebra, both the Weyl ten­sor of conformal curvature and the conharmonic curvature ten­sor vanish. We also establish that the vanishing of the scalar cur­vature of a six-dimensional planar submanifold of Cayley algebra implies the vanishing of both the Weyl tensor of conformal cur­va­ture and the conharmonic curvature tensor.

Two problems related to geodesic points of six-dimensional planar submanifolds of the octave algebra are formulated. Solu­tions, or significant advances in solving of these problems, appear en­tirely possible.

In this paper, we propose a classification of finite-dimensional real Lie algebras that admit subalgebras of codimension two, but no subalgebra of codimension one, which leads to a description of ho­mogeneous spaces without isotropy subgroups of codimension one. This study expands upon the existing body of research on co­dimension-one subalgebras, aiming to extend the analysis to a new setting. The result of this expansion is a comprehensive description of the relevant families. It is demonstrated that a such Lie algebra  contains an ideal , containing the radical of , such that  be­longs to a list , consisting of one compact and two non compact Lie algebras.

This article proposes a method for constructing the second fundamental form of a  surface of positive curvature. It is well known that constructing the second fundamental form of a surface requires at least a twice continuously differentiable parametriza­tion. This paper uses a probabilistic approach to this problem, in­troducing the concept of the second Itô derivative along the direc­tion of Brownian motion. In more detail, we define the second de­rivative “without the second derivative”. For this, we use the Itô for­mula, restricted to special random processes. Using the pro­posed construction, we can write Weingarten's equations for sur­fa­ces of almost bounded curvature (the paper has been accepted for publication in the Ufa Mathematical Journal) and determine the mean curvature almost surely over the entire surface pointwise. To so­me extent, this paper can be viewed as a generalization of I. Ya. Bakelman’s work “Differential Geometry of Smooth Irregu­lar Surfaces” (see [5]). In the author’s opinion, further reduction of sur­face smoothness is difficult to achieve using analytical methods (the technique developed in [5]), and new research methods are nee­ded. This paper proposes one such method: a stochastic frame­work for studying low-smoothness surfaces. Previously, a slightly dif­ferent approach was used, based on transition functions and tran­sition densities of random processes; see, for example, [6] (which also contains references to earlier works).

Using the group-theoretic method of differential geometric re­search by G. F. Laptev, as well as the methodology of scientific re­search on differential geometry by V. S. Malakhovsky, the study of complexes (three-parameter families) of ellipsoids with special pro­perties of differentiable maps associated with them in a three-di­mensional affine space. The complex of ellipsoids is considered as an image of the corresponding mapping. A subclass of previous­ly studied ellipsoid complexes is investigated when the difference indicatrix of the second and third coordinate vectors is a straight line parallel to the first coordinate vector. Interesting geo­metric properties of the studied manifold are found, which make it possi­ble to construct it, that is, to find its integral representation.

Let  be the tangent bundle of a complete Riemannian manifold  equipped with the Sasaki metric  and  be a horizontal-harmonic map from  to  If  for at least one positive number , then  is a harmonic map from  to

An RR-polyhedron is a closed convex polyhedron in E3 whose face set can be partitioned into two non-empty disjoint sets: the set of faces forming isolated faceted stars of symmetric rhombic verti­ces and the set of regular faces. If the regular faces of such a poly­hedron are of the same type, then it belongs to the first type; if they are different, then it belongs to the second type of RR-polyhedron. In the present paper, it is proved that there exist only six non-composite RR-polyhedrons of the first type with conditional edges. Two of the found RR-polyhedrons can be partitioned by planes passing through the conditional edges of the rhombic faces into regular parts, and one of the RR-polyhedrons can be partitioned in­to two rhombic pyramids.

In a projective space a distribution of the second kind inde­pendently of the geometry of distributions of the first kind is stu­died. A distribution of the second kind is understood as an n-di­mensional manifold of multidimensional planes obtained by as­signing to each point of the space a plane not passing through that point. In the principal bundle associated with the distribution, a connection is given. An equipment of the distribution of the sec­ond kind is constructed; it is a field of planes that, together with the generating point and the generating plane, forms a three-term composition in the sense of Norden. It is proved that the distribu­tion and its equipment induce connections of two types in the as­sociated bundle. Conditions for the coincidence of these connec­tions are found.