Isotropic manifold

（重定向自Isotropic space）

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 $v,w\in T_pM$ , there is an isometry $\varphi$ of $M$ with $\varphi(p)=p$ and $\varphi_\ast(v)=w$ . Every complete isotropic manifold is homogeneous, i.e. for any $p,q\in M$ there is an isometry $\varphi$ of $M$ with $\varphi(p)=q.$ 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 $\gamma'(1)$ to $-\gamma'(1).$

单词	Isotropic space
释义	Isotropic space 英语百科 Isotropic manifold （重定向自Isotropic space） 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 $v,w\in T_pM$ , there is an isometry $\varphi$ of $M$ with $\varphi(p)=p$ and $\varphi_\ast(v)=w$ . Every complete isotropic manifold is homogeneous, i.e. for any $p,q\in M$ there is an isometry $\varphi$ of $M$ with $\varphi(p)=q.$ 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 $\gamma'(1)$ to $-\gamma'(1).$
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