A characterization of complex projective space up to biholomorphic isometry

Abstract

Obata [3] gave a characterization up to isometry of the standard sphere sn in terms of the Hessian operator on a complete Riemannian manifold. With the convention Hess u = Vgrad u, his result says that if M is a complete Riemannian manifold which admits a nondegenerate function u such that Hess u = -u- Id then M is isometric to the standard sphere. Obata goes on to prove related results in conformal geometry which take advantage of the existence of a function whose Hessian has a special form. Other authors have also obtained strong geometric properties of a Riemannian manifold by exploiting the existence of a function u with Hess u = f - Id for some function f. In particular, this last equation implies that M is a warped product. For a proof and related results, see, for example, Osgood and Stowe [5]. In the complex case, a characterization of cn up to isometry was given by Stoll [6] via the complex Monge-Amp~re operator. Stoll's result says that if M is a complex manifold which admits a strictly plurisubharmonic exhaustion r: M --> [0, ~) such that (dd c log r) n -- 0, then (M, r) = (C", Iz12). In other words M, with hermitian metric given by the Kahler form ddCr, is biholomorphically isometric to C". Obata also showed that a complete, connected and simply connected Kahler manifold is isometric to the complex projective space ]?n if and only if it admits a solution to a certain linear system of third order differential equations [4]. Blair [1] subsequently showed that in some cases this characterization of ]?n follows from a corresponding result for Riemannian manifolds and indicated that one would not expect a characterization of ]?n by a Hessian equation analogous to that which Obata used to characterize S n. In this paper we give a complex analog of Obata's theorem [3]. We characterize complex projective space up to biholomorphic isometry by the existence of a solution to a system of second order equations. Since ]?n with the Fubini-Study metric is not a warped product, there does not exist a nontrivial function u on ]?n whose Hessian is a multiple of the identity. However, ]?n with one point deleted is the hyperplane section bundle over I? n- 1 and the fibers of this bundle are totally geodesic complex lines. Thus, there is a relationship between the natural metric structure and the line bundle structure of ~n. This relationship provides the motivation for the construction of a function u on ]?n