Institute for Low Temperature Physics and Engineering
The Department of Geometry was organized in 1960. Since that moment up to 2000 POGORELOV A.V. headed this Department. In 2000-2014 the Department of Geometry was headed by Professor Yu.A. Aminov. In 2014 the Department of Geometry was reorganized and included into a newly organized Department of Differential Equations and Geometry.
One of the burning and fruitful lines of modern geometry is a geometry in the large. This division of the geometry science treats of the surfaces and other geometrical objects in the large, but not locally, viz in the small. Intensive proceeding of this area of geometry was started in early 1930. when Dr. A.D. Alexandrov had proposed a highly efficient method to study a general convex surface by a special approaching of its intrinsic metrics via polytopes. Using this method, he solved the well-known Weyl problem on arbitrary convex metrics. The well-known Alexandrov's theorem on gluing made it possible reduce the classical problem of convex surfaces bending to the problems of rigidity for general convex surfaces and regularity of a convex surface with a regular metrics. A.V. Pogorelov has solved the above problems in the late 1940s and in early 1950s. Subsequently, Dr. A.V. Pogorelov has solved the problem of infinitesimal bendings of general convex surfaces in Euclidean and constant-curvature spaces, constructed the theory of surfaces of limited extrinsic curvature, advanced geometric theory of Monge- Ampere differential equation of elliptic type, which can present themselves during the investigation of a variety of problems of geometry in the large. The results obtained by A.V. Pogorelov contributed significantly to geometry in the large in the late 1950s. But they set many new problems which are now attached by the Department of Geometry under the head and direct participation of academician A.V. Pogorelov.
The principal trends of the work of the Department are:
1) geometry in the large of surfaces and metrics;
2) geometric theory of stability of the shells or application of geometry in the large to the mechanics of thin-wall shells.
In the field of geometry in the large the following fundamental results were obtained by workers of the Department:
- Problems of isometric immersions, regularity, rigidity and infinitesimal rigidity of locally convex surfaces of the Riemannian space were solved (A.V. Pogorelov, 1960-1968).
- The fourth Hilbert problem was completely solved (A.V. Pogorelov, 1973-1974).
- The problem of improper affine hyperspheres was solved, a regular solution of the Minkowsky multidimensional problem was obtained (A.V. Pogorelov, 1972-1975).
- A metric theory of shortest and geodesic lines on general hypersurfaces and in multidimensional polyhedral metrics of non- negative curvature was developed (A.D. Milka, 1969-1979).
- The following problems were solved: rectifiability of the spherical image of the shortest line on a convex surface; smoothness and strict convexity of a convex hypersurface with restrictions on the specific curvature; classification of points of a shortest line as points of a convex hypersurface (A.D. Milka, 1969 - 1979).
- The well-known A.D. Alexandrov's uniqueness theorem was dimension-generalized for analytical closed surfaces with non-go in curvature indicatrix (A.I. Medianik, 1966-1970).
- Existence theorems for convex surfaces having an edge with the prescribed functions of the principal curvature radii as functions of the normal were proved (A.I. Medianik, 1972-1975).
- General estimates were obtained for the external diameter of multidimensional submanifolds in the Euclidean space via internal magnitudes and the modulus of the mean curvature vector (Yu.A. Aminov, 1972-1975).
- The theorem of instability of a closed minimal surface homeomorphic to the sphere or the torus in a positive curved Riemannian space was proved (Yu.A. Aminov, 1975-1976).
- The problems of continuous bendings of convex surfaces with a convex or concave edge in the Euclidean and pseudo-Euclidean spaces and rigidity of the general closed convex surfaces in the Lobachewsky and de Sitter spaces (A.D. Milka, 1982-1986).
- A new combinatorial class of uniqueness theorems for convex polyhedra generalizing the well-known Cauchy and Minkowsky theorems was distinguished and investigated (A.D. Milka, 1986).
- The whole list of combinatorial types of Euclidean polytopes with equiangular vertices was found and finiteness of the set of such polytopes in the spherical space was proved (A.D. Milka, A.M. Gurin, 1980-1987).
- A geometric theory of the multidimensional analogue of the Monge-Ampere equation of the elliptic type was constructed (A.V. Pogorelov, 1983-1988).
- A new type of theorem on the existence and uniqueness of a regular closed convex surface whose sum of reciprocal quantities of the Gaussian and mean curvatures is a function of the normal with sufficiently small oscillations satisfying the closeness condition was proved (A.I. Medianik, 1989-1991).
- The theory of isometric immersions of the n-dimensional Lobachewsky space into the (2n-1)-dimensional Euclidean space was constructed (Yu.A. Aminov, 1977-1995).
- The theory of the Grassmann image of submanifold was developed. Theorems on the reconstruction of submanifolds from a given Grassmann image were proved in the following cases: for submanifolds with one- or two-dimensional Grassmann image, for three-dimensional submanifolds of great codimension, for submanifolds with flat normal connection, for submanifolds represented explicitly, for two-dimensional closed surfaces in four-dimensional Euclidean space with particular closed Grassmann image etc. Generic and particular (conformal, isometric, areal) deformations preserving the Grassmann image are described for two-dimensional surfaces in the four-dimensional Euclidean space. Completely solved the problem of reconstruction of a spherical submanifold from its given Grassmann image. (Yu.A. Aminov,V.A. Gorkaviy,1980-2012).
- In connection with a well-known problem on the existence of closed bendable surface in 3-dimensional space the class of surfaces for which each component of the position vector is a trigonometric polinomial in two variables was considered.Two theorems on the non-bendability of surfaces in this class are proved, and an expression for the volume of the domain bounded by such the surface is established (Yu.A. Aminov, 1990).
- A theory of H. Busemann's regular G-spaces was constructed (A.V. Pogorelov, 1990-1993).
- It was proved that to within combinatorial equivalence there exists only a restricted number of various infinite convex irre- ducible polyhedra with equiangular faces, besides three infinite series: a truncated cone with a finite or infinite base and a right prism with infinite bases (A.M. Gurin,A.I. Medianik, 1991-1992).
- It was proved that in 3-dimensional Euclidean, spherical and hyperbolic spaces a closed convex surface of internal diameter d, with the additional condition of central symmetry, may be subdivided into four part, each having an internal diameter smaller then d at its surface (A.D. Milka, 1992).
- A new type of continuous bendings of polyhedra was considered, so-called linear bendings, i.e. piecewise linear isometric deformations. Some classes of linear bendings and related discret isometries of regular polyhedra were studied. A special linear bending of the cube and related realizations, which are nonrigid in the infinitesimal, second order ones included, were investigated. The theorems were established that relate linear bendings to the finite group theory, the modular function theory and the classical theory and its applications. Particular linear bendings which increase the volume were constructed for right regular pyramids, cylinders and convex regular polyhedra (A.D. Milka, 1993- 1994, A.D. Milka Ė V.A. Gorkavyy, 2008-2012).
- The existence of a smooth closed surface inside a tetrahedron which contacts all its faces and has a prescribed constant mean curvature on the part lying strictly inside the tetrahedron was proved (A.V. Pogorelov, 1994);
- It was proved that there exists no local isometric immersion of the Lobachewsky 3-dimensional space into the 5-dimensional Euclidean space with a constant internal curvature of the Grassmanian image (Yu.A. Aminov, 1994).
- The well-known Hopf problem consists in proving impossibility of the existence of a positive curvature metrics on the topological product of two 2-dimensional spheres. The metrics on this product was considered which is equal to the sum of two 2-dimensional metrics, conformally equivalent to metrics of a constant positive curvature with the conformity coefficients depending on points of the space. It was proved for simple conditions for this coefficients that there exist points and tangent planes therein for which the curvature of the space metrics is zero (Yu.A. Aminov, 1994).
- It was proved that two isometric spacelike identically oriented convex polyhedra homeomorphic to the circle with a non-negative turning edge admit continuous bending into one another in the class of convex spacelike polyhedra (A.D. Milka, 1995).
- The theorem on bendability of a circle-homeomorphic convex surface with a positive geodesic curvature of the edge into a convex surface with the same spherical image, with a given point at the boundary being the image by the isometry of the respective point, was proved (A.V. Pogorelov, 1995).
- Linear bendings were constructed for star-like bi-pyramids of Alexandrov-Vladimirova. A mathematical model describing a non-stability of physical models of particular polyhedra, model flexors, was proposed (A.D. Milka, 1997-2004).
- The existence of analogs of the Bianci-Baecklund transformations for two-dimensional surfaces in many-dimensional spaces of constant curvature Ňn, Sn, Hn and in Riemannian products Sn x R1, Hn x R1 is proved. Pseudo-spherical surfaces in Ň4, S4, H4, that admit Bianci transformations are completely described. The existence of pseudo-spherical submanifolds in Ňn, Sn, Hn, which do not admit Bianci-Baecklund transformations is alsoproved (Yu. A. Aminov, V.A. Gorkavyy, E.N. Neverzhitska, 1999-2012) .
- A general expression for the Riemann tensor is derived for submanifolds in Riemannian spaces represented implicitly by systems of equations(Yu. A. Aminov, 1999-2002).
- Some expressions are derived for the volume V of a coordinate parallelepiped constructed on asymptotic lines of a pseudo-spherical submanifold realized via an isometric immersion of a domain of the Lobachevski space L3 into the Euclidean space E5. An estimate from above for V is proved. A conjecture of J.D. Moore about isometric immersions of domains of The Lobachevskis pace is disproved (Yu. A. Aminov, 2001-2002).
- An estimate from above is derived for the radius of a ball in E2n-1, which may contain a family of n-dimensional submanifolds of constant negative curvature K £ -1 making part of an orthogonal coordinate system in E2n-1 (Yu. A. Aminov, 2004-2005).
- A multi-dimensional generalization of a theorem of Shwartz on the stability of minimal surfaces in E3 included into a regular family of minimal surfaces is proved/ The generalization concerns minimal hypersurfaces in Riemannian spaces Rn and for two-dimensional surfaces in Riemannian spaces R4 (Yu. A. Aminov, 2004-2005).
- Necessary and sufficient conditions for the existence a semi-cyrculant matrix of Hadamard of degree 4n are derived. Geometric conditions are related to the existence of antipodal n-angles inscribed into a right (2n-1)-angle. Analytic conditions deals with the real solvability of some particular non-homogeneous algebraic system as well as with some particular cubic hypersurface in (2n-1)-dimensional projective space (A.I. Medianik, 2003-2013).
- Theorems on the existence of a right hyper-symplex inscribed into a (4n-1)-dimensional cube are proved (A.I. Medianik, 2004-2013).
- Conformal and isometric deformations preserving the Grassmann image are described for two-dimensional surfaces in the multi-dimensional Minkowski space. Two particular classes of light-like surfaces, which admit non-trivial conformal or isometric deformations preserving the Grassmann image, like minimal surfaces, are distinguished. An effective method for constructing concrete examples of the mentioned light-like surfaces is proposed (V.A. Gorkavyy, 2004-2010).
- Necessary and sufficient conditions for a closed revolution surfaces in E3 to admit a short deformation increasing the volume are derived (V.A. Gorkavyy, 2010-2012).
- A particular class of rigid packings with equal balls in the three-dimensional Euclidean space, which includes Hylbertís packing, is described (A.M. Gurin, 1995).
- A geometric analysis of metallic glass and Bernall liquid is fulfilled (A.M. Gurin, 1989-2005).
- A problem of H.Stuck on hyper-foliation of non-negative sectional curvature in odd-dimensonal spheres is partially solved, some topological obstructions are discovered (D.V. Bolotov, 2002).
- A macroscopic dimension is investigated for Riemannian manifolds. A conjecture of M. Gromov about the macroscopic dimension is confirmed for three-dimensional compact manifolds (D.V. Bolotov, 2003).
- The problem of existence of totally geodesic distributions on compact Thurston manifolds is solved (D.V. Bolotov, 2004).
- The geometry of wave functions of electrons is analyzed. A wave function is interpreted as a three-dimensional submanifold in the five-dimensional Euclidean space. Expressions for the metric form, Riemannian curvature and curvature of normal connection are derived. The behavior of sectional and scalar curvatures at extremal points of submanifolds is analyzed, some relations to quantum numbers are discovered (Yu.A. Aminov, 2013-2014).
- The existence of polynomial solutions of the Monge-Ampere equation with polynomial right-hand side is analyzed. Some theorems on existence or non-existence of polynomial solutions are proved (Yu. A. Aminov, 2013-2014).
The profound results of A.V. Pogorelov in the theory of finite and infinitesimal bendings of convex surfaces obtained in late 40s and 50s were applied by the Department of Geometry to mechanics of thin-wall shells. Namely:
- A new geometric method for investigation of non-linear problems of deformation of convex shells,their stability and postcritical behaviour was proposed and developed, viz the GEOMETRIC THEORY OF STABILITY OF SHELLS (A.V. Pogorelov, 1960-1967, 1978-1979). This method provides solution of some well-known as well as new problems of stability of shells. The final results could be presented, as a rule, in a closed form. Discontinuous infinitesimal and finite non-regular bendings of cylindrical, conical, convex general developable and general strictly convex surfaces reproducing the shape of the respective shell for the initial and significant postcritical deformations were constructed. Various fashions of loading, including standard distributed and local loads as well as combined loading, were considered. Linearly and non-linearly elastic and elastico-plastic, isotropic and anisotropic, shallow and non-shallow, both closed and those with an edge shells (perfect or imperfect in form), with rigid fixing or free supporting along the edge were investigated. The upper and lower critical loads, loading diagrams, i.e. the load-sagging dependences, were determined (A.V. Pogorelov, V.I. Babenko, V.V. Mikhaylov, 1960-1994);
- A new original method was proposed and developed for obtaining experimental samples of precision shapes by coper deposition in vacuum (A.V. Pogorelov, 1960-1964). This permitted a series of experiments to be carried out which were first to corroborate some principal findings of theoretical studies of stability of shells produced by both classical methods and the new geometric one. In particular, experimental substantiation was provided of the formulae for the upper critical Lorentz-Timoshenko loads for cylindric shells under axial compression and the Pogorelov ones for shallow rigidly fixed strictly convex shells under external pressure (the validity of the Zoelly formula for shallow spherical shells) as well as the formulae of the geometric theory of stability shells in the case of the lower critical loads for cylindric shells under axial compession, external pressure, torsion and for spherical shells under external pressure (A.V. Pogorelov, V.I. Babenko,M.M. Pugolovok,V.M. Prichko, 1960-1994);
- The phenomenon of geometrically non-linear localization of initial postcritical deformations of a shell was predicted, discavered and studied; the related problems of stability of convex shells (developable shells in particular) were considered, including problems of substantiation of the geometric method of study of loss of stability of strictly shells (V.I. Babenko, 1972-1986);
- Boundary-valued problems on local loss of stability of isotropic spherical,anisotropic general strictly convex and convex developable shells, both homogeneous and three-layered were studied by the asymptotic method with various loading and fixing fashions (V.I. Babenko, 1973-1986);
- The complete pressure-sagging diagram at the vertex was obtained experimentally for first time which includes a postcritical (descending) branch for shallow, rigidly fixed, spherical and elliptically paraboloid shells.It was found that the loading diagram obtained experimentally coincides with the calculated one (obtained by the classical methods) everywhere except a certain neigbourhood of the upper critical load, which has not yet been explained in terms of the Kirchoff-Love theory of shells (V.I. Babenko, V.M. Prichko, V.Sh. Avedyan, 1981-1994);
- Varimodulus shells with Rabotnov-Lomakin's elasticity potential were considered,and elasticity relations were obtained for them. The edge effect and the local loss of general strictly convex and convex developable shells were studied by the asymptotic method for various loading and fixing fashions manners. The loading diagram for spherical shells under external pressure was studied numerically (V.I. Babenko, A.D. Senenko, 1983-1995);
- Subcritical non-linear deformation and loss of stability of rigidly fixed shallow double curvature shells having the shape of convex toroidal segments that are circular in ground under external pressure was studied numerically (V.I. Babenko, V.Sh. Avedyan, 1993).
Another application of geometry in the large is to be noted which has been developed at the Geometry Department since the early 90s, namely that using computers:
- In the area of interactive computer graphic a method of geometric modelling of lines and surfaces was proposed which is efficiently realized and provides better potentialities than wellknown Besier and B-Spline methods (A.D. Milka, A.V. Gubar, 1993- 1994);
- A computer algorithm was developed for construction of order 4n half-circulant Hadamard matricies, which is equivalent to construction of a regular (4n-1)-dimensional symplex inscribed into the cube of the same dimension (A.I. Medianik, 1994-1995). In contrast to the well-known Williamson method, the new one gives also an infinite series of Hadamard matricies (while the number 2n-1 is a prime number).