Complementary event

Template:Distinguish2 In mathematics, an isotropic manifold is a manifold in which the geometry does not depend on directions. Formally, we say that a Riemannian manifold $(M,g)$ is isotropic if for any point $p\in M$ and unit vectors $v,w\in T_{p}M$, there is an isometry $\phi$ of $M$ with $\phi (p)=p$ and ${\phi }_{\ast }(v)=w$. Every isotropic manifold is homogeneous, i.e. for any $p,q\in M$ there is an isometry $\phi$ of $M$ with $\phi (p)=q.$ This can be seen by considering a geodesic $\gamma :[0,2]\to M$ from $p$ to $q$ and taking the isometry which fixes $\gamma (1)$ and maps ${\gamma }^{\prime }(1)$ to $-{\gamma }^{\prime }(1).$

Examples

The simply-connected space forms (the n-sphere, hyperbolic space, and ${\mathbb{ℝ}}^n$) are isotropic. It is not true in general that any constant curvature manifold is isotropic; for example, the flat torus $T={\mathbb{ℝ}}^2/{\mathbb{\mathbb{Z}}}^2$ is not isotropic. This can be seen by noting that any isometry of $T$ which fixes a point $p\in T$ must lift to an isometry of ${\mathbb{ℝ}}^2$ which fixes a point and preserves ${\mathbb{\mathbb{Z}}}^2$; thus the group of isometries of $T$ which fix $p$ is discrete. Moreover, it can be seen 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 ${\mathbb{ℂ}\mathbb{\mathbb{P}}}^n$ ($n>1$) equipped with the Fubini-Study metric.

Further examples of isotropic manifolds are given by the rank one symmetric spaces, including the projective spaces ${\mathbb{ℝ}\mathbb{\mathbb{P}}}^n$, ${\mathbb{ℂ}\mathbb{\mathbb{P}}}^n$, ${\mathbb{ℍ}\mathbb{\mathbb{P}}}^n$, and ${\mathbb{𝕆}\mathbb{\mathbb{P}}}^2$, as well as their noncompact hyperbolic analogues.

A manifold can be homogeneous but not isotropic, such as $\mathbb{ℝ}\times S^2$ with the product metric.

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