Global structure Hopf hypersurfaces in symmetric spaces of rank one
by
A. A. Borisenko
Kharkov National University, Ukraine

Let M be a hypersurface of Riemannian manifold [`M], [`R] the curvature tensor of [`M] and \xi a unit normal vector of M at p in M. The normal Jacobi operator

K\xi: = R   ( . ,  \xi) \xi in End (TpM)

of M (with respect to \xi) takes the curvature of [`M] at p into consideration, and the shape operator A<sub>\xi</sub> of M (with respect to \xi) describes the curvature of M at p relative to the ambient space [`M]. We call M curvature-adapted to M, if the normal Jacobi operator K and the shape operator A of M are simultaneously diagonalizable, that is, if K_\xi o A_\xi=A_\xi o K_\xi for all unit normal vectors \xi of M.

In complex space forms C[`M] (c) and quaternionic space forms H[`M] (c) curvature adapted hypersurfaces are Hopf hypersurfaces. If \xi is normal to a hypersurface M in C[`M] (c) then the vector J\xi is a tangent vector to M, where J is the complex structure in C[`M] (c). For Hopf hypersurfaces the vector J\xi is a principal direction at every point of M [1]. The analogical is true for Hopf hypersurfaces in quaternionic space forms H[`M] (c) [2].

Theorem 1 [3] Let M be a C^2n-1 regular compact generic immersed orientable Hopf hypersurface in the complex projective space C Pⁿ (n >= 2). Then M is a tube over an irreducible algebraic variety.

Corollary Let M be a C^2n-1 regular connected compact embedded Hopf hypersurface in the complex projective space CPⁿ  (n >= 2). Then M is a tube over an irreducible algebraic variety.

Theorem 2 [3] Let M be a C^2n-1 regular connected compact generic immersed orientable Hopf hypersurface in the complex projective space C Pⁿ (n >= 2) contained in a geodesic ball of radius R < [(\pi)/2]. Then M is a geodesic hypersphere.

Let C Hⁿ be the complex hyperbolic space of constant holomorphic curvature -4. For C Hⁿ we prove the following theorem.

Theorem 3 [3] Let M be a connected compact generic immersed orientable C^2n-1 regular Hopf hypersurface in the complex hyperbolic space CHⁿ (n >= 2). Then the Hopf hypersurface M is a geodesic hypersphere.

Theorem 4 Let M be a Hopf hypersurface in complex hyperbolic space C Hⁿ. Suppose that principal curvature at the direction J\xi is equal 2 at some point of M. Then the hypersurface M is a horosphere.

The analogical theorem is true for Hopf hypersurfaces in quaternionic hyperbolic space and it follows.

Theorem 5 Hopf hypersurfaces in quaternionic space forms H[`M] (c)  (c =/= 0) have constant principal curvatures.

I. In quaternionic projective space H Pⁿ  (c > 0) the Hopf hypersurfaces are:

1) a tube of some radius r in ]0,  \pi/2[ around the cononically (totally geodesic) embedded quaternionic projective space H P^k for some k in { 0,  ... ,  n-1} ;

2) a tube of some radius r in ]0,  \pi/4[ around the cononically (totally geodesic) embedded complex projective space C Pⁿ.

II. In quaternionic hyperbolic space H Hⁿ  (c < 0) the Hopf hypersurfaces are:

1) a tube of some radius r in R₊ around the cononically (totally geodesic) embedded quaternionic hyperbolic space H H^k for some k in { 0,  ... ,  n-1} ;

2) a tube of some radius r in R₊ around the cononically (totally geodesic) embedded complex hyperbolic space C Hⁿ.

3) a horosphere in HHⁿ.

References

[1] T.E. Cecil, P.J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982)481-499.

[2] J. Berndt, Real hypersurfaces in quaternionic space forms, J. Reine. Angev. Math. 419 (1991) 9-26.

[3] A.A. Borisenko, Compact Hopf hyperfaces, Dokl. of the National Academie Nauk of Ukraine 8 (1999) 13-17.

Date received: July 17, 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 # cafh-41.