John Carlos Baez on Nostr: I'm getting really interested in symmetric spaces. These are manifolds M where for ...

I'm getting really interested in symmetric spaces. These are manifolds M where for each point p there's an operation called 'reflection about p',

Rₚ: M → M

which leaves p fixed:

Rₚ (p) = p

gets you back where you started if you do it twice:

RₚRₚ(m) = m for all m ∈ M

and most crucially, acts as -1 on the tangent space of p:

dRₚ = -1

Often we also demand that M is a Riemannian manifold and the operations Rₚ preserve the metric.

For example, a sphere is a symmetric space. For each point on a sphere, there's an obvious way to "reflect about that point" which sends each tangent vector at the point (in blue) to its negative (in red).

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