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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:

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