[Math] Isotropic Manifolds

differential-geometryriemannian-geometry

Recall that a Riemannian manifold $(M,g)$ is isotropic if for any $p\in M$ and any unit vectors $v,w\in T_pM$ there is an isometry $f:M\to M$ such that $f_\ast(v)=w.$ Recall also that $(M,g)$ is homogeneous if for any $p,q\in m$ there is an isometry $f:M\to M$ such that $f(p)=q.$

I would appreciate answers to any of the following questions:

  1. I know that the simply connected constant curvature spaces are isotropic. What are some other examples of isotropic manifolds?

  2. I suspect that the flat torus is not isotropic and that the product metric on $M\times N$ is not isotropic if $M$ and $N$ are not isometric. Are these true?

  3. Is there some way to relate transitivity of the holonomy group on the unit sphere in $T_pM$ to isotropy?

  4. Are isotropic manifolds and symmetric spaces related?

  5. What are some results connecting homogeneity and isotropy? Homogeneous isotropic manifolds are important in relativity; are these classified, at least in dimension 3? When can we conclude that a homogeneous manifold is isotropic? When can we conclude that an isotropic manifold is homogeneous?

  6. What are some references for reading about isotropic manifolds? I've tried using google and a few Riemannian geometry and general relativity books (Petersen, Wald, Choquet-Bruhat) but I haven't been able to find very much.

Best Answer

Here's a proof that all isotropic manifolds are homogeneous. Given any $p$ and $q$ in $M$, let $\gamma:[0,2]\rightarrow M$ be a minimizing geodesic from $p$ to $q$. Set $r = \gamma(1)$. So, following $\gamma'(1)$ along for one unit of time lands you at $q$ while following it backwards for one unit o ftime lands you at $p$.

By assumption, there is an isometry $f$ for which $d_r f$ maps $\gamma'(1) \in T_r M$ to $-\gamma'(1)$. Then, by uniqueness of geodesics, we have \begin{align*} f(q) &= f(\exp_r ( \gamma'(1)))\\ &= \exp_r(d_r f \, \gamma'(1)) \\ &= \exp_r(-\gamma'(1)) \\ &= p.\end{align*}

As other have pointed out, among, say, compact simply connected homogeneous spaces, isotropic spaces are very rare. In fact, given such a homogeneous space $G/H$ (where we assume wlog $H$ and $G$ share no common normal subgroups of positive dimension), this space is isotropic iff the induced action of $H$ on $T_{eH} G/H$ is transitive on the unit sphere. Under these assumptions, one can prove, for example, that the universal cover of $H$ has at most two factors. (In fact, those $H$ which act effectively and transitively on a sphere have been completely classified.)

I don't know of a classification of when $G/H$ has $H$ acting transitively on the unit sphere, but beyond $\mathbb{C}P^n$, it also happens for $\mathbb{H}P^n$ and $\mathbb{O}P^2$ (the compact, rank one, symmetric spaces). I don't know any other examples of homogeneous spaces for which $H$ acts transitively on the sphere.

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