A Riemannian manifold is, roughly speaking, a space in which we can measure lengths and angles. The most symmetrical of these are called "symmetric spaces". In 2 dimensions there are 3 kinds, but in higher dimensions there are more.

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An "isometry" of a Riemannian manifold M is a one-one and onto function f: M → M that preserves distances (and thus angles). Isometries form a group. You should think of this as the group of symmetries of M.

For M to be very symmetrical, we want this group to be big!

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The group of isometries of a Riemannian manifold is a manifold in its own right! So it has a dimension.

The isometry group of the plane, sphere or saddle (hyperbolic plane) is 3-dimensional. This is biggest possible for the isometry group of a 2d Riemannian manifold!

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For an n-dimensional Riemannian manifold, how big can the dimension of its isometry group be? n(n+1)/2. And this happens in just 3 cases:

n-dimensional Euclidean space,
the n-dimensional sphere, and
n-dimensional hyperbolic space (a "hypersaddle").

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So, whatever definition of "highly symmetrical Riemannian manifold" we choose, these 3 cases deserve to be included.

Another great bunch of examples come from "Lie groups": manifolds that are also groups, such that multiplication is a smooth function.

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The best Lie groups are the "compact" ones. These can be made into Riemannian manifolds in such a way that left/right multiplication by any element is an isometry! These have finite volume.

We can completely classify compact Lie groups, and study them endlessly.

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So, any decent definition of "symmetric space" should include Euclidean spaces, spheres, hyperbolic spaces and compact Lie groups - like the rotation groups SO(n), or the unitary groups U(n).

And there's a very nice definition that includes all these - and more!

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A Riemannian manifold M is a "symmetric space" if it's connected and for each point x there's an isometry f: M → M called "reflection around x" that maps x to itself and reverses the direction of any tangent vector at x:

f(x) = x

and

dfₓ = -1

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For example, take Euclidean space. For any point x, "reflection around x" maps each point x + v to x - v.

So it maps x to itself, and reverses directions!

To understand symmetric spaces better, it's good to mentally visualize "reflection around x" for a sphere.

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We can completely classify *compact* symmetric spaces - and spend the rest of our life happily studying them.

Besides the compact Lie groups, there are 7 infinite families and 12 exceptions, which are all connected to the octonions:

https://en.wikipedia.org/wiki/Symmetric_space#Classification_result

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Symmetric spaces are great if you like geometry, because here's an almost equivalent definition: they are the Riemannian manifolds whose curvature tensor is preserved by parallel translation!

(Some fine print is required for a complete match of definitions.)

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Symmetric space are also great if you like algebra!

Exactly as Lie groups can be studied using Lie algebras, symmetric spaces can be studied using "Z/2-graded Lie algebras", or equivalently "Lie triple systems".

I explained all that here:

(11/n) https://twitter.com/johncarlosbaez/status/1275802637732990977

Even better, the 7+3 = 10 infinite series of compact symmetric spaces (the seven I mentioned plus the three infinite series of compact Lie groups) are fundamental in condensed matter physics! The "10-fold way" classifies states of matter:

http://math.ucr.edu/home/baez/tenfold.html

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So, our search for the most symmetrical spaces leads us to a meeting-ground of algebra and geometry that generalizes the theory of Lie groups and Lie algebras and has surprising applications to physics! What more could you want?

Oh yeah....

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You might want to *learn* this stuff. Wikipedia is good:

https://en.wikipedia.org/wiki/Symmetric_space

These notes have lots of examples:

http://myweb.rz.uni-augsburg.de/~eschenbu/symspace.pdf

Then try Helgason's "Differential Geometry, Lie Groups and Symmetric Spaces" - I learned Lie groups from him, and this book.

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Then, to really sink into the glorious details of symmetric spaces, I recommend Arthur Besse's book "Einstein Manifolds".

Besse is a relative of the famous Nicolas Bourbaki. His book has lots of great tables. Lots of fun to browse!

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