Isometries of Hermitian symmetric spaces

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Proposition 1 (Chen-Nagano 1978) If M is a compact Riemannian symmetric space of rank k, then any meridian M − ⊂ M has the same rank k. Proposition 2 A meridian M − in an extrinsically symmetric space M is extrinsically symmetric. 26
Proposition 3 Let M be a compact Riemannian symmetric space which satisfies that every polar is a single point. Then M is the Riemannian product of round spheres and possibly a torus. Proposition 4 Let P ⊂ R n be a full extrinsically symmetric flat torus. Then P splits extrinsically as a product of round circles, P = S 1 r1 × · · · × S1 rn ⊂ R2n . 27
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 and 22: Lemma (I f)∗(Jo) ∈ {±Jo}. Proo
Page 23 and 24: Remark It is a known fact that ever
Page 25: 4. Another result Theorem 2 (Loos 1
Page 29: Further, a meridian M − of X is e

Isometries of Hermitian symmetric spaces

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