Analytic Extension of Riemannian Analytic Manifolds and Local Isometries

This article deals with a locally given Riemannian analytic manifold. One of the main tasks is to define its regular analytic extension in order to generalize the notion of completeness. Such extension is studied for metrics whose Lie algebra of all Killing vector fields has no center. The generalization of completeness for an arbitrary metric is given, too. Another task is to analyze the possibility of extending local isometry to isometry of some manifold. It can be done for metrics whose Lie algebra of all Killing vector fields has no center. For such metrics there exists a manifold on which any Killing vector field generates one parameter group of isometries. We prove the following almost necessary condition under which Lie algebra of all Killing vector fields generates a group of isometries on some manifold. Let g be Lie algebra of all Killing vector fields on Riemannian analytic manifold, h⊂g is its stationary subalgebra, z⊂g is its center and [g,g] is commutant. G is Lie group generated by g and is subgroup generated by h⊂g. If h∩(z+[g;g])=h∩[g;g], then H is closed in G.

Stationary Subalgebras in General Position for Tensor Product

The paper studies stationary subalgebras in general position of compact linear groups. We prove that, except for several specific cases, a stationary subalgebra in general position of a tensor product of real or complex compact group representations acts as a scalar on all tensor factors but possibly one. In the real case, it means that this stationary subalgebra in general position is contained in one of the direct summand subalgebras. We used the following concepts to solve this problem: conventional linear algebra arguments; theory of Lie groups, Lie algebras and their representations; and methods similar to those of solving similar problems for complex reductive linear groups.

RATIONAL SOLUTIONS OF THE CYBE AND LOCALLY TRANSITIVE ACTIONS ON ISOTROPIC GRASSMANNIANS

This paper continues the investigation of the theory of rational solutions of the CYBE for o(n) from the point of view of orders in the corresponding loop algebra, as it was developed in [8]. As suggested by [8], in the case of "singular vertices", we use the list of connected irreducible subgroups of SO(n) locally transitive on the Grassmann manifolds [Formula: see text] of isotropic k-dimensional subspaces in ℂn obtained in [11]. New arguments based on the analysis of the structure of the stationary subalgebra of a generic point allow us to construct several rational solutions in o(7), o(8) and o(12).