Abstract
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.
Highlights
For a long time, the “curvilinearity” of our space was scientifically substantiated
We proved that the extension of the isometry ψ along all possible curves to Vε yields a one-to-one mapping defined on the whole V, it can be proved that the extension of φ along all possible curves to M gives an isometric embedding φ : M → N
We present the proof for the case when the Lie algebra of all Killing vector fields has no center
Summary
The “curvilinearity” of our space was scientifically substantiated. The geometry of our space does not obey the laws of Euclidean geometry, but is determined by the general concept of the Riemannian metric. Let M be a regular pseudocomplete Riemannian analytic manifold whose Lie algebra of all vector fields has no center, S be the set of fixed points of all local isometries of the manifold M preserving f0 be orientation and Killing vector fields, M0 be a quasicomplete manifold locally isometric to M, M connected covering of the manifold M0. Let M be pseudocomplete analytic Riemannian manifold, z⊥ the distribution of tangent vectors perpendicular to the center z of the algebra of all Killing vector fields, S be the set of fixed points of local isometries preserving the orientation and all Killing vector fields.
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