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In a Riemannian symmetric space $Q$, it is well known that the existence of a totally geodesic submanifold at a point $p \in Q$ is equivalent to the existence of a Lie triple system at $p$, i.e., a subspace $V$ of $T_{p}Q$ such that $[u, [v, w]] \in V$ for all $u,v,w \in V$; see for example Theorem 7.2 in Helgason, Differential geometry, Lie groups, and symmetric spaces.

As far as I understand, the proof carries over directly to the case where the ambient manifold is a semi-Riemannian symmetric space.

Question. Does anyone know where in the literature I can find a statement/proof of this more general result?

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  • $\begingroup$ Isn't that for all torsion-free affine connections? $\endgroup$
    – alvarezpaiva
    Commented Dec 7, 2021 at 8:34

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Try pages 71 and 72 of Cartan's "La géométrie des groupes de transformations" https://eudml.org/doc/235668

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