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 formula_1 is isotropic if for any point formula_2 and unit vectors formula_3, there is an isometry formula_4 of formula_5 with formula_6 and formula_7. Every connected isotropic manifold is homogeneous, i.e. for any formula_8 there is an isometry formula_4 of formula_5 with formula_11 This can be seen by considering a geodesic formula_12 from formula_13 to formula_14 and taking the isometry which fixes formula_15 and maps formula_16 to formula_17 Examples. The simply-connected space forms (the  n-sphere, hyperbolic space, and formula_18) are isotropic. It is not true in general that any constant curvature manifold is isotropic; for example, the flat torus formula_19 is not isotropic. This can be seen by noting that any isometry of formula_20 which fixes a point formula_21 must lift to an isometry of formula_22 which fixes a point and preserves formula_23; thus the group of isometries of formula_20 which fix formula_13 is discrete. Moreover, it can be seen in a same way that no oriented surface with constant curvature and negative Euler characteristic is isotropic. Moreover, there are isotropic manifolds which  do not have constant curvature, such as the complex projective space formula_26 (formula_27) equipped with the Fubini-Study metric. Indeed, all constant curvature manifolds have their universal cover to be either a sphere, or a hyperbolic space, or formula_18, but formula_26 is simply-connected but not a sphere (for formula_27), as can be seen for example from homotopy group calculations from long exact sequence of the fibration formula_31. Further examples of isotropic manifolds are given by the rank one symmetric spaces, including the projective spaces formula_32, formula_26, formula_34, and formula_35, as well as their noncompact hyperbolic analogues. A manifold can  be homogeneous but not isotropic, such as the flat torus formula_20 or formula_37 with the product metric.