Isotropic manifold

In mathematics, an isotropic manifold is a manifold in which the geometry does not depend on directions. Formally, we say that a Riemannian manifold is isotropic if for any point and unit vectors , there is an isometry of with and . Every complete isotropic manifold is homogeneous, i.e. for any there is an isometry of with This can be seen by considering a geodesic from to and taking the isometry which fixes and maps to