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A unitary-symmetric Kahlerian manifold is a Kahlerian version of a rotationallysymmetric (Riemannian) manifold (cf. Choi [3], Greene-Wu [5]). Precisely, a Kahlerian manifold (M, g, J) of complex dimension n is unitary-symmetric at a point p of M if the linear isotropy group at p of the automorphism group of (M, g, J) is the unitary group U(ri). Of course, the complex space form is unitary-symmetric at every point. The firstpurpose of this paper is to give one characterization of a connected, simply-connected, complete, unitary-symmetric Kahlerian manifold. If M is compact, then the tangential cut locus Cp of p is spherical. Hence (M, g, J) is a Blaschke manifold at p and has a SZ/-structure (cf. Besse [1]). Then the second purpose is to give a sufficientcondition in order that a connected, compact, unitary-symmetric Kahlerian manifold has a SCp-structure (Theorem D) (see Besse [1, p. 181]). On the other hand, Greene-Wu [5, p. 85] introduced the notion of a Hermitian rotationallysymmetric manifold of complex dimension 1 and Shiga [12] studied a Kahlerian model, which is by defintion a Kahlerian manifold with a pole p such that the linear isotropy group at p of the isometry group is U(n). Note that their manifolds are unitary-symmetric Kahlerian manifolds. The unitarysymmetric condition is a fairly strong one, because the result of Kaup [8, Folgerung 1.10] implies that a connected, unitary-symmetric Kahlerian manifold is biholomorphic to one of the complex space forms. But there exist many complete unitary-symmetric Kahlerian metrics, which are not isometric to them (see Mori-Watanabe [10]). Throughout this paper, (M, g, J) is assumed to be a connected, complete Kahlerian manifold of complex dimension n^l. To state our results, we prepapre the following. By Q we denote the Kahlerian form of (M, g, J). We frequently identify the tangent space TP(M) at a point p of M with the complex number n-space Cn. Let expp be the exponential map of TJM)

Unitary-symmetric Kählerian manifolds and pointed Blaschke manifolds