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In mathematics, an isotropic manifold is a manifold in which the geometry does not depend on directions. Formally, we say that a Riemannian manifold {\displaystyle } is isotropic if for any point p ∈ M {\displaystyle p\in M} and unit vectors v , w ∈ T p M {\displaystyle v,w\in T_{p}M} , there is an isometry φ {\displaystyle \varphi } of M {\displaystyle M} with φ = p {\displaystyle \varphi =p} and φ ∗ = w {\displaystyle \varphi _{\ast }=w}. Every connected isotropic manifold is homogeneous, i.e. for any p , q ∈ M {\displaystyle p,q\in M} there is an isometry φ {\displaystyle \varphi } of M {\displaystyle M} with φ = q . {\displaystyle \varphi =q.} This can be seen by considering a geodesic γ : → M {\displaystyle \gamma :\to M} from p {\displaystyle p} to q {\displaystyle q} and taking the isometry which fixes γ {\displaystyle \gamma } and maps γ ′ {\displaystyle \gamma '} to − γ ′ . {\displaystyle -\gamma '.}