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Vol 37, No 1 (2016)
- Year: 2016
- Articles: 9
- URL: https://journals.rcsi.science/1995-0802/issue/view/12333
Article
3-webs with singularities
Abstract
A 3-web with singularities is an ordered collection of three one-dimensional distributions L1, L2, L3 on a 2-dimensional manifold M. The subset Σ ⊂ M where these distributions are not pairwise transversal is called the singularity set. Under some conditions on Σ we find the differential invariants of the 3-web with singularities at the points of Σ and give examples of calculation of these invariants.
1-20
A note on the relation between joint and differential invariants
Abstract
We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie’s finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential invariants, and we expose a general method that allows to compute differential invariants from joint invariants.
21-35
Differential contra algebraic invariants: Applications to classical algebraic problems
Abstract
In this paper we discuss an approach to the study of orbits of actions of semisimple Lie groups in their irreducible complex representations,which is based on differential invariants on the one hand, and on geometry of reductive homogeneous spaces on the other hand. According to the Borel–Weil–Bott theorem, every irreducible representation of semisimple Lie group is isomorphic to the action of this group on the module of holomorphic sections of some one–dimensional bundle over homogeneous space. Using this, we give a complete description of the structure of the field of differential invariants for this action and obtain a criterion which separates regular orbits.
36-49
Contact structures as Dirac structures and their associated Poisson algebras
Abstract
We give a description of contact structures on smoothmanifolds as Dirac structures, and we compute their associated Poisson algebras of admissible functions. We also revisit the notion of moment map associated to smooth actions on Lie groups on contact manifolds, we show that a contact moment map induces a moment map in the Dirac sense as defined in [3], and this map induces a natural morphism of Lie algebras of vector fields, admissible functions and infinitesimal symmetries.
50-59
A natural geometric construction underlying a class of Lax pairs
Abstract
In the framework of the theory of differential coverings [2], we discuss a general geometric construction that serves the base for the so-called Lax pairs containing differentiation with respect to the spectral parameter [4]. Such kind of objects arise, for example, when studying integrability properties of equations like the Gibbons–Tsarev one [1].
60-65
Complexes of differential forms associated with a normalized manifold over the algebra of dual numbers
Abstract
We construct some complexes of differential forms on a smooth manifold MnD over the algebra of dual numbers D on the base of a decomposition of the tensor product TMnD⊗ℝD into the Whitney sum of two subbundles. It is shown that these complexes can be obtained as restrictions of some complexes of holomorphic (D-smooth) forms defined on the tangent bundle TMnD. For holomorphic fiber bundles over MnD, we introduce complexes of D-valued forms holomorphic along the fibers and express in terms of cohomology classes of such complexes the obstructions to existence of holomorphic connections in holomorphic principal bundles.
66-74
Ricci flow on the barrel S1 × [−1, 1]
Abstract
We study a boundary value problem for the Ricci flow on a surface with the topological type of the cylinder S1 × [−1, 1], which we refer to as barrel, and we give estimates on the rates of convergence for the total scalar curvature and the area of the surface at time t. We present a family of examples for which our theorems apply.
75-79
On the equivalence problem of generalized Abel ODEs under the action of the linear transformations pseudogroup
Abstract
In the present paper we establish the necessary and sufficient conditions for two generalized Abel differential equations to be locally equivalent under the action of the pseudogroup of linear transformations of the form {x ↦ f(x), y ↦ g(x) · y + h(x)}. These conditions are formulated in terms of differential invariants.
80-86
Geometric structures on solutions of equations of adiabatic gas motion
Abstract
In this paper we show that characteristic covectors of equations of n-dimensional adiabatic gas motion, n = 1, 2, 3, generate a geometric structure on every their solution. This structure consists of a hyperplane and a non degenerate cone in each cotangent space to a solution so that the hyperplane and the cone intersect only at the zero point. We investigate differential invariants of this structure. In particular, we find a natural linear connection on every solution. A torsion tensor of this connection is trivial for n = 1. For n = 2, 3, this tensor is not trivial in general. For n = 1, we calculate solutions having the linear connection with zero curvature tensor. For n = 2, 3, we calculate solutions with zero torsion tensor.
87-100