×

The intersection of two real forms in Hermitian symmetric spaces of compact type. II. (English) Zbl 1335.53068

In this paper the authors continue the investigation of intersections of real forms (i.e., fixed point sets of anti-holomorphic isometries) in compact Hermitian symmetric spaces started in [the authors, ibid. 64, No. 4, 1297–1332 (2012; Zbl 1263.53047); correction ibid. 67, No. 3, 1161–1168 (2015); ibid. 67, No. 3, 1161–1168 (2015; Zbl 1332.53070)]. There they obtain a precise description of such intersections in the irreducible case. Here they deal with the reducible case. They prove that a real form in a compact Hermitian symmetric space \(M\) is a product of real forms in irreducible factors of \(M\) and diagonal real forms determined by antiholomorphic isometries between irreducible factors of \(M\) (given an antiholomorphic isometric map between compact Hermitian symmetric spaces \(\tau: M_1\to M_2\), then the diagonal real form of \(M_1\times M_2\) determined by \(\tau\) is given by \(D_\tau(M_1)=\{(x,\tau(x) | x\in M_1\}\)).
Next they show that the intersection of two real forms can be reduced to either the intersection of two real forms in an irreducible factor or to the intersection of two diagonal real forms in the product of two copies of an irreducible factor. The former case was already investigated in [Zbl 1263.53047, loc. cit.]. For the latter case, they obtain the following result.
Theorem. Let \(M_1\), \(M_2\) be irreducible Hermitian symmetric spaces of compact type which are holomorphically isometric. We take two anti-holomorphic isometric maps \(\tau_1 : M_1 \to M_2\) and \(\tau_2 : M_2 \to M_1\). We assume that the intersection of \(D_{\tau_1} (M_1)\) and \(D_{\tau_2^{-1}} (M_1)\) is discrete. Then we have the following.
(1)
If \(M_1 = Q_{2m}({\mathbb C})\), with \(m\geq 2\), and \(\tau_2\tau_1\) does not belong to \(A_0(M_1)\), then \[ \#(D_{\tau_1} (M_1) \cap D_{\tau_2^{-1}} (M_1)) = 2m < 2m+2 = \#_2 M_1. \]
(2)
If \(M_1 = G_m({\mathbb C}^{2m})\), with \(m \geq 2\), and \(\tau_2\tau_1\) does not belong to \(A_0(M_1)\), then \[ \#(D_{\tau_1} (M_1) \cap D_{\tau_2^{-1}} (M_1)) = 2^m < {2m\choose m} = \#_2 M_1. \]
(3)
Otherwise, \(D_{\tau_1} (M_1) \cap D_{\tau_2^{-1}} (M_1)\) is a great antipodal set of \(D_{\tau_1} (M_1) \) and \(D_{\tau_2^{-1}} (M_1)\), thus \[ \#(D_{\tau_1} (M_1) \cap D_{\tau_2^{-1}} (M_1)) =\#_2M_1. \]
\(A_0 (M)\) is the identity component of the group of holomorphic isometries of \(M_1\).

MSC:

53C35 Differential geometry of symmetric spaces
53C40 Global submanifolds
53D12 Lagrangian submanifolds; Maslov index
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] D. S. P. Leung, Reflective submanifolds. IV, Classification of real forms of Hermitian symmetric spaces, J. Differential Geom., 14 (1979), 179-185. · Zbl 0458.53033
[2] O. Loos, Symmetric Spaces. II: Compact spaces and classification, W. A. Benjamin, 1969. · Zbl 0175.48601
[3] S. Murakami, On the automorphisms of a real semi-simple Lie algebra, J. Math. Soc. Japan, 4 (1952), 103-133. · Zbl 0047.03501
[4] M. Takeuchi, On the fundamental group and the group of isometries of a symmetric space, J. Fac. Sci. Univ. Tokyo Sect. I, 10 (1964), 88-123. · Zbl 0196.54605
[5] M. Takeuchi, Stability of certain minimal submanifolds of compact Hermitian symmetric spaces, Tohoku Math. J. (2), 36 (1984), 293-314. · Zbl 0528.53047
[6] M. S. Tanaka and H. Tasaki, The intersection of two real forms in Hermitian symmetric spaces of compact type, J. Math. Soc. Japan, 64 (2012), 1297-1332. · Zbl 1263.53047
[7] M. S. Tanaka and H. Tasaki, Correction to: “The intersection of two real forms in Hermitian symmetric spaces of compact type”, to appear in J. Math. Soc. Japan. · Zbl 1332.53070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.