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 [math]\displaystyle{ (M,g) }[/math] is isotropic if for any point [math]\displaystyle{ p\in M }[/math] and unit vectors [math]\displaystyle{ v,w\in T_pM }[/math], there is an isometry [math]\displaystyle{ \varphi }[/math] of [math]\displaystyle{ M }[/math] with [math]\displaystyle{ \varphi(p)=p }[/math] and [math]\displaystyle{ \varphi_\ast(v)=w }[/math]. Every connected isotropic manifold is homogeneous, i.e. for any [math]\displaystyle{ p,q\in M }[/math] there is an isometry [math]\displaystyle{ \varphi }[/math] of [math]\displaystyle{ M }[/math] with [math]\displaystyle{ \varphi(p)=q. }[/math] This can be seen by considering a geodesic [math]\displaystyle{ \gamma:[0,2]\to M }[/math] from [math]\displaystyle{ p }[/math] to [math]\displaystyle{ q }[/math] and taking the isometry which fixes [math]\displaystyle{ \gamma(1) }[/math] and maps [math]\displaystyle{ \gamma'(1) }[/math] to [math]\displaystyle{ -\gamma'(1). }[/math]
Examples
The simply-connected space forms (the n-sphere, hyperbolic space, and [math]\displaystyle{ \mathbb{R}^n }[/math]) are isotropic. It is not true in general that any constant curvature manifold is isotropic; for example, the flat torus [math]\displaystyle{ T=\mathbb{R}^2/\mathbb{Z}^2 }[/math] is not isotropic. This can be seen by noting that any isometry of [math]\displaystyle{ T }[/math] which fixes a point [math]\displaystyle{ p\in T }[/math] must lift to an isometry of [math]\displaystyle{ \mathbb{R}^2 }[/math] which fixes a point and preserves [math]\displaystyle{ \mathbb{Z}^2 }[/math]; thus the group of isometries of [math]\displaystyle{ T }[/math] which fix [math]\displaystyle{ p }[/math] 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 [math]\displaystyle{ \mathbb{CP}^n }[/math] ([math]\displaystyle{ n\gt 1 }[/math]) equipped with the Fubini-Study metric. Indeed, the universal cover of any constant-curvature manifold is either a sphere, or a hyperbolic space, or [math]\displaystyle{ \mathbb{R}^n }[/math]. But [math]\displaystyle{ \mathbb{CP}^n }[/math] is simply-connected yet not a sphere (for [math]\displaystyle{ n\gt 1 }[/math]), as can be seen for example from homotopy group calculations from long exact sequence of the fibration [math]\displaystyle{ U(1)\to S^{2n+1}\to \mathbb{CP}^n }[/math].
Further examples of isotropic manifolds are given by the rank one symmetric spaces, including the projective spaces [math]\displaystyle{ \mathbb{RP}^n }[/math], [math]\displaystyle{ \mathbb{CP}^n }[/math], [math]\displaystyle{ \mathbb{HP}^n }[/math], and [math]\displaystyle{ \mathbb{OP}^2 }[/math], as well as their noncompact hyperbolic analogues.
A manifold can be homogeneous but not isotropic, such as the flat torus [math]\displaystyle{ T }[/math] or [math]\displaystyle{ \mathbb{R}\times S^2 }[/math] with the product metric.
See also
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