# Homogenous but not isotropic manifolds

$$(M,g)$$ is homogeneous if $$\forall a,b\in M$$ there is an isometry $$f:M\to M$$ such that $$f(a)=b$$

$$(M, g)$$ is called isotropic if $$\forall p\in M$$ $$x,y\in T_pM$$ of unit length, there is an isometry $$f:M\to M$$ s.t. $$f_\ast(x)=y$$

I know that Every connected isotropic manifold is homogeneous (given 2 points, take the middle of the geodesics connecting them)

Now, are there homogenous manifolds that are not isotropic? Please, explain in detail if so.

A cylinder $$S^1\times \mathbb{R}$$ will do the job ! In general, isometries of a product Riemannian manifolds are not the "product of isometries of the factors", but in this simple case I hope we can agree it's true: an isometry of the cylinder is of the type $$(x,y)\mapsto (r(x),\pm y+a),$$ where $$r\in O(2)$$ and $$a\in \mathbb{R}$$. If you want a rigorous proof of this fact, check this answer. Clearly the isometries of the cylinder map tangent vectors like $$(v,0)$$ in tangent vectors of the same kind (the last coordinate stays equal to $$0$$), so the cylinder is not isotropic, but it's homogeneous.

I want to add an interesting curiosity. Isotropic connected riemannian manifolds are completely classified up to isometry and are:

• Euclidean spaces $$\mathbb{E}^n$$,
• Spheres $$\mathbb{S}^n$$,
• Hyperbolic (real) spaces $$\mathbb{H}^n$$,
• Hyperbolic complex spaces $$\mathbb{H}^n\mathbb{C}$$,
• Hyperbolic quaternionic spaces $$\mathbb{H}^n\mathbb{H}$$,
• Hyperbolic octonionic plane $$\mathbb{H}^2\mathbb{O}$$,
• Projective real spaces $$\mathbb{P}^n\mathbb{R}$$,
• Projective complex spaces $$\mathbb{P}^n\mathbb{C}$$,
• Projective quaternionic spaces $$\mathbb{P}^n\mathbb{H}$$,
• Projective octonionic plane $$\mathbb{P}^2\mathbb{O}$$.

While homogeneous Riemannian manifolds are much more! They are precisely the manifolds $$G/H$$ where $$G$$ is a Lie group and $$H\leq G$$ is a compact subgroup (basically a complete classification is near impossible).

• Great! Thank you very much
– Jack
Commented May 22 at 8:44
• The last sentence is wrong, it should be "...quotient manifolds $G/H$ where $G$ is a Lie group and $H\le G$ is a compact subgroup..." Commented May 22 at 13:54
• @MoisheKohan Thank you for the correction! Commented May 22 at 14:57
• Still wrong: $G/H$ is not a group, just a manifold. Commented May 22 at 14:59
• @MoisheKohan Thank you again! Commented May 22 at 14:59