Isometries of Hermitian symmetric spaces

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Let F := ±(I f)∗ if (I f)∗(Jo) = ±Jo, respectively. To prove f = F | P , we show ι ◦ f = F ◦ ι. Since we have (ι ◦ f)(o) = (F ◦ ι)(o) and ι ◦ f ◦ γ = F ◦ ι ◦ γ for arbitrary geodesic γ starting from o, we obtain the conclusion. 22
Remark It is a known fact that every reflective submanifold of a semisimple Hermitian symmetric space is a symmetric R-space. Because we know every reflective submanifold of a semisimple Hermitian symmetric space is either a complex submanifold or a real form from the next proposition. 23
Page 1 and 2: Isometries of Hermitian symmetric s
Page 3 and 4: 1. Introduction A Riemannian symmet
Page 5 and 6: M : a Riemannian manifold τ : an i
Page 7 and 8: 2. Extrinsically symmetric spaces M
Page 9 and 10: P = G/K : a Riemannian symmetric sp
Page 11 and 12: I(M) : the group of isometries of M
Page 13 and 14: 3. Isometries of Hermitian symmetri
Page 15 and 16: Consider the map ι : P → g, p
Page 17 and 18: Proof: P = P1 × · · · × Pr : a
Page 19 and 20: Let f ∈ I(P ). Let m be the midpo
Page 21: Lemma (I f)∗(Jo) ∈ {±Jo}. Proo
Page 25 and 26: 4. Another result Theorem 2 (Loos 1
Page 27 and 28: Proposition 3 Let M be a compact Ri
Page 29: Further, a meridian M − of X is e

Isometries of Hermitian symmetric spaces

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