Geometry of isometry groups in notions of the geometry of Riemannian manifolds

In this paper, we will have the different glance to the geometry of the isometry group of a Riemannian manifold. In fact, the (Lie) group of isometries of a Riemannian manifold M,g, I M, acts on M properly and this action is specified by giving the orbits of points of M as an immersed submanifold of it. These submanifolds intersect the other submanifolds, i.e., slices at the fixed points of the action transversely. In this way, M by a suitable fibration can be expressed in the letters of the quotients of I M and its slices as the fiber. Also, there is the explicit corresponding relation between their Lie algebras. On the other hand, another glance to tangent bundle TM as an associated bundle to the frame bundle M , LM, results to the one-to-one correspondence between the special GLm,R -equivariant functions and vector field M . This correspondence via a Lie algebra homomorphism between the Lie algebras of TM, LI M and M , translates the main tools of the geometry of TM to the ones of M . We reach to these fundamental and applied conclusions through this paper.

کلیدواژه ها:

Isometries ، Proper actions ، Slices ، Associated bundles ، Equivariant maps ، Lie algebra homomorphisms