Isometries of Hermitian symmetric spaces

2. Extrinsically symmetric spaces
M ⊂ R n : a submanifold of Euclidean space
M is called extrinsically symmetric if it is preserved by
the reflection ρp at the affine normal space p + (TpM) ⊥
for every p ∈ M. Here ρp : R n → R n is the affine isometry
with ρp(p) = p, ρp| TpM = −Id and ρp| (TpM) ⊥ = Id.
Every extrinsically symmetric submanifold is a Rieman-
nian symmetric space w.r.t. the induced metric. In fact,
the geodesic symmetry sp at p ∈ M is given by sp = ρp| M.
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