Tubes and the Geometry of Kähler manifolds
by
Christina C. Beneki
University of Patras, Greece
Coauthors: Andreas Arvanitoyeorgos (The British Council-Athens, Greece and the University of Essex, UK)

Poster

One of the central problems in Differential Geometry is the characterization of a Riemannian manifold by using properties of a certain submanifold. This problem has been studied by several authors (e.g Blair, Ledger, Vanhecke-Willlmore, Papantoniou) by the use of small geodesic spheres in the case of a Kähler manifold with constant holomorphic sectional curvature, Grassman manifolds, Riemannian manifold with constant sectional curvature or quaternionic Kähler manifold with constant Q-sectional curvature. After the work of Gray on tubes, several characterizations of the above manifolds by using small tubes about a geodesic, have been achieved. Recent works toward this direction are the ones of Djoric and Gillard. In our work we characterize a Kähler manifold with constant holomorphic sectional curvature by the use of small tubular hypersurfaces about a certain submanifold. We prove the following.

Theorem 1 Let (M, g, J) be a connected 2n-dimensional (n > 1) Kähler manifold of constant holomorphic sectional curvature, and let P be a q-dimensional submanifold of M obtained by exp_m(V \cap B_\e (0)), where V is a q-dimensional subspace of the tangent space T_mM of M at a point m in M, and B_\e (0) an open ball centered at the origin of T_mM. Let \gbe a geodesic meeting P orthogonally at m=\g (0) in P, let g '(t) \equiv N, and let {\e₁, ..., \e_2n} be a parallel frame field along \gchosen in such a way, so that \e_q+1 \equiv \gamma'(t) \equiv N,  \e_q \equiv JN, and \e_q+2 \equiv \xi. Then every sufficiently small tube about P is a quasi-umbilical hypersurface of M with respect to the plane spanned by JN and \xi, and in particular the space operator of each such tube has a parallel eigenspace along \gof dimension 2n-(q+2). By using a theorem of Kosmanec we also prove the converse of the above theorem

Theorem 2 Let (M, g, J) be a connected 2n-dimensional (n > 1) Kähler manifold, P a q-dimensional submanifold of M as defined in theorem 1, and let \gbe a geodesic meeting P orthogonally at m=\g (0) in P. Assume that every sufficiently small tube about P is a quasi-umbilical hypersurface of M with respect to the plane spanned by JN and \xi, and in particular the corresponding shape operator has a parallel eigenspace along \gof dimension 2n-(q+2). Then M is of constant holomorphic sectional curvature.

We will also mention some recent work on the characterization of quaternionic Kähler manifolds with constant Q-sectional curvature by using small tubes about a geodesic.

Date received: May 9, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-42.