Isotropic Manifolds

Question

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.

Answer 1

There is a class of spaces variously known as (1) rank one symmetric spaces and/or (2) two-point homogeneous spaces. These are isotropic (and of course homogeneous). A key example to think about are complex projective spaces and complex hyperbolic spaces. Using some of these terms as keywords you should be able to make more progress. Certainly Helgason's book is a must.

Answer 2

I'll leave a more detailed, systematic answer to an expert, but "morally speaking" it's harder for a Riemannian manifold to be isotropic than to be homogeneous; a few examples may indicate why:

• The Fubini-Study metric on $\mathbf{CP}^n$ is isotropic, but does not have constant (real) sectional curvature.

• A point in a flat torus has finite stabilizer in the isometry group, so is far from isotropic. (An isometry of a $2$-torus lifts to a lattice-preserving isometry of $\mathbf{R}^2$.)

• Flat $\mathbf{R}^n$ is isotropic, but has trivial holonomy. A constant-curvature oriented surface of genus $g \geq 2$ has discrete isometry group but holonomy $SO(2)$.

• I don't have these at hand to verify their aptness, but you might try Einstein Manifolds by Besse, The Shape of Space by Weeks (a delightful read in any event), Foundations of Differential Geometry by Kobayashi and Nomizu, or Spaces of Constant Curvature by Wolf.

Answer 3

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.