Locally Isometric Riemannian Analytic Spaces

Classes of locally isometric Riemannian analytic manifolds are studied. A generalization of the concept of completeness is given. We consider the Lie algebra ???? of all Killing vector fields of a Riemannian analytic manifold, its stationary subalgebra ???? the simply connected Lie group ???? corresponding to the Lie algebra ????, and the subgroup ???? corresponding to the Lie subalgebra ????. In the absence of a center in the algebra ???? the concept of a quasi-complete (compressed) manifold is introduced. An oriented Riemannian analytic manifold whose vector field algebra has zero center is said to be quasi-complete if it is non-extendable and does not admit non-trivial orientation-preserving and all Killing vector fields local isometries to itself. The main property of such a manifold is that it is unique in the class of all locally isometric Riemannian analytic manifolds, and any locally given isometry of this manifold ???? into itself can be analytically extended to an isometry ????: ???? ≈ ????. For an arbitrary class of locally isometric Riemannian analytic manifolds, a definition of a pseudocomplete manifold is given, which is complete if a complete manifold exists in the given class. A Riemannian analytic simply connected manifold M is called pseudocomplete if it has the following properties. ???? is non-extendable. There is no locally isometric covering map f; M→N, where N is a simply connected Riemannian analytic manifold and f (M) is an open subset of N not equal to N.