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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
Full Text:
##### References:
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