## Acerca de la Geometría de Lobachevski by A. S. Smogorzhevski By A. S. Smogorzhevski

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3. 2 Pseudo-Hermitian symmetric spaces. We have seen in the previous section that in order to classify indecomposable pseudo-Hermitian symmetric spaces one would have to classify the objects (O1)–(O3) for ˆl being a complex grading. This task is of a similar complexity as the classification of all symmetric spaces discussed in Section 3. 1/-equivariance and making some aspects of the theory simpler then for general symmetric spaces. 3, Table 1) admit a Kähler structure. The classiﬁcation problem for pseudo-Riemannian symmetric spaces 33 We treat pseudo-Hermitian symmetric spaces together with para-Hermitian ones because of the similarity of their behaviour.

L / Š R and D . L/ej0 (7) (8) 26 I. Kath and M. Olbrich for L 2 l, i D 1; : : : ; p and j D 1; : : : ; q. l; Âl ; a WD . R; id/-module for which the 1-eigenspace of the involution is positive definite is of the kind defined above. 0; 0/g and indecomposability Since l is one-dimensional we have HQ and admissibility conditions are easy to handle. 2 (Cahen–Wallach ). p; q; ; /, p; q 0, p C q > 0, . ; / 2 Mp;q , where ˇ 0< 1Ä Ä p; 0 < 1 Ä ˇ Mp;q WD . ; / 2 Rp ˚ Rq ˇ 1 D 1 if p > 0; 1 D 1 else Ä q ; : Cahen and Wallach  constructed explicit models for all simply connected Lorentzian symmetric spaces.

4. are C; C; S 2 Š CP 1 , and H 2 , respectively. The complex gradings of l correspond to the natural complex structures of these spaces. Note that the data defined in 4. do not depend on the choice of the square root of z. l; ˆl /-modules in cases 3. and 4. 3. 1 differs slightly from the corresponding statement in . 2; 2q/ that admit the structure of a pseudo-Hermitian symmetric triple, whereas here we have determined isomorphism classes of pseudo-Hermitian symmetric triples. Comparing both results we find that all of these symmetric triples admit exactly one complex grading (up to isomorphism).