Generalization of the notion of completeness of a Riemannian analytic manifold

In this paper, we discuss the concept of an analytic continuation of a local Riemannian metric. We propose a generalization of the notion of completeness realized as an analytic continuation of an arbitrary Riemannian metric. Various Riemannian metrics are studied, primarily those related to the structure of the Lie algebra $\mathfrak{g}$ of all Killing vector fields for a local metric. We introduce the notion of a quasicomplete manifold, which possesses the property of extendability of all local isometries to isometries of the whole manifold. A classification of pseudocomplete manifolds of small dimensions is obtained. We present conditions for the Lie algebra of all Killing vector fields $\mathfrak{g}$ and its stationary subalgebra $\mathfrak{h}$ of a locally homogeneous pseudo-Riemannian manifold under which a locally homogeneous manifold can be analytically extended to a homogeneous manifold.