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In learning about symmetric spaces, I have come across many examples ($\mathbb{R}^n, S^n$ and $\mathbb{P}^n$ to name a few), but I haven't come across many counterexamples.

I am aware that symmetric spaces are complete, so that would imply that any example of a noncomplete space, such as $\{\mathbb{R}-\{0\}\}$ should work.

I am also aware that simply connected symmetric spaces are uniquely determined by there curvature at a point, which leads me to believe that the "egg" surface, shown below, which is a $C^2$ surface given by the union of a sphere and the surface of revolution of a polynomial cannot be a symmetric space either.

the egg

Are there any other examples of (preferably visualisable) nonsymmetric spaces?

Flumpo
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  • Just take the graph of some sufficiently random smooth function of two variables. (For instance, $f(x,y)=x^2+y^2$.) This graph, with the induced metric, has nonzero curvature, hence, will not be a symmetric space. – Moishe Kohan Feb 04 '21 at 22:54
  • Thank you, this seems to do the trick. – Flumpo Feb 06 '21 at 16:27

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You have to start out with a homogeneous space $G/H$, so the group of Riemannian isometries has to act transitively, first of all. Yes, the curvature tensor has to be parallel, so in the case of a surface, the manifold must have constant curvature. So any surface in $\Bbb R^3$ with non-constant curvature cannot be a symmetric space.

Grassmannians are the first "interesting" example of symmetric spaces. These are not spaces of constant (sectional) curvature. $G(k,n)$ is the space of $k$-dimensional subspaces of $\Bbb R^n$. (You can also do oriented subspaces, and you can do complex subspaces of $\Bbb C^n$, as well.)

The easiest example I know of a homogeneous space that fails to be (locally) symmetric is a flag space like $$X = \{(\ell,H)\in G(1,3)\times G(2,3): \ell\subset H\},$$ the collection of ordered pairs of lines through the origin in $\Bbb R^3$ and planes through the origin with the property that the line is contained in the plane. It is a circle bundle over $\Bbb RP^2$. This example generalizes to all dimensions, considering subspaces contained in larger-dimensional subspaces.

Ted Shifrin
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  • Thank you for your answer. Are there any simple examples of locally symmetric spaces which fail at being globally symmetric? It seems like the upper half plane and other restrictions of globally symmetric spaces should give some examples and the disjoint union of multiple symmetric spaces should give others, though I'm not sure if there are locally symmetric spaces which do not derive from globally symmetric ones. – Flumpo Feb 06 '21 at 16:27
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    Yes, there are, but this is a somewhat sophisticated question. Indeed, the definition of (locally) symmetric spaces in terms of reversal of geodesics does not presuppose the homogeneity that I did. See, for example, this and this. – Ted Shifrin Feb 06 '21 at 18:47
  • I think these might be a tad more complex than I had in mind.Also, thinking over it a little more, it seems I was wrong in thinking the disjoint union of symmetric spaces is symmetric. Am I correct that the upper half plane $x^n>0$ is locally symmetric but not globally symmetric? – Flumpo Feb 06 '21 at 22:39
  • With the hyperbolic metric it is globally symmetric. It is complete and simply connected. As those answers suggest, you'll need some quotients by strange discrete groups, I think. – Ted Shifrin Feb 06 '21 at 23:10