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A class of symmetric spaces. (English) Zbl 0697.53047
Let (M,\(\nabla)\) be a connected \(C^{\infty}\) manifold with a linear torsion free connection \(\nabla\) on its tangent bundle. The author calls (M,\(\nabla)\) projectively symmetric if for every point x of M there is an involutive projective transformation of M fixing x and whose differential at s is -Id. The assignment of the symmetry \(s_ x\) at each point x of M must not be continuous.
In this paper the author gives necessary and sufficient conditions for a projectively symmetric and projectively homogeneous space to be inessential (i.e. projectively equivalent to an affine symmetric space). For complete Riemannian manifolds (M,g) of dimension n (n\(\geq 3)\) which are projectively symmetric and projective homogeneous, the author proves that such spaces are either inessential or isometric to the sphere \(S^ n(r)\) or to the projective space \(S^ n(r)/-Id\) with some choice of symmetries.
Reviewer: H.Özekes

MSC:
53C35 Differential geometry of symmetric spaces
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References:
[1] EISENHART (L.P.) . - Non-Riemannian Geometry . - Amer. Soc. Colloq. Publ., t. 8, 1927 .
[2] KOBAYASHI (S.) and NOMIZU (K.) . - Foundations of Differential Geometry . - New York, J. WILEY and SONS, vol. 1 and vol. 2, 1963 , 1969 . · Zbl 0119.37502
[3] KOBAYASHI (S.) . - Transformation Groups in Differential Geometry . Springer Verlag, 1980 .
[4] KOWALSKI (O.) . - Generalized Symmetric Spaces , [Lect. Notes in Math.], Springer Verlag, 1980 . MR 83d:53036 | Zbl 0431.53042 · Zbl 0431.53042
[5] LEDGER (A.J.) and OBATA (M.) . - Affine and Riemannian s-Manifolds , J. Differential Geom., t. 2, 1968 , p. 451-459. MR 39 #6206 | Zbl 0177.24602 · Zbl 0177.24602
[6] NAGANO (T.) . - The projective transformation of a space of parallel Ricci tensor , Kodai Math. J., t. 11, 1959 , p. 131-138. Article | MR 22 #216 | Zbl 0097.37503 · Zbl 0097.37503 · doi:10.2996/kmj/1138844182 · minidml.mathdoc.fr
[7] NOMIZU (K.) and PINKALL (U.) . - On the geometry of affine immersions , Math. Z., t. 195, 1987 , p. 165-178. MR 88e:53089 | Zbl 0629.53012 · Zbl 0629.53012 · doi:10.1007/BF01166455 · eudml:183684
[8] PODESTA (F.) . - Projectively Symmetric Spaces , to appear in Ann. di Mat. Pura e Appl.. Zbl 0697.53048 · Zbl 0697.53048 · doi:10.1007/BF01790357
[9] SINJUKOV (S.) . - Geodesic mappings of Riemannian spaces on symmetric Riemannian spaces , Dokl. Akad. Naukk (Russian), t. XC-VIII, 1954 , p. 21-23. MR 16,515f | Zbl 0056.15301 · Zbl 0056.15301
[10] SOLODOVNIKOV (S.) . - The group of projective transformations in a complete analytic Riemannian space , Soviet Math. Dokl., t. 10, 1969 , p. 750-753. Zbl 0188.54003 · Zbl 0188.54003
[11] VEBLEN (O.) . - Generalized projective Geometry , J. London Math. Soc., t. 4, 1929 , p. 140-160. JFM 55.0413.02 · JFM 55.0413.02
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