Thomas E. Cecil

Focal sets and real hypersurfaces in complex projective space

Let M be a real submanifold of CPm, and let J denote the complex structure. We begin by finding a formula for the location of the focal points of M in terms of its second fundamental form. This takes a particularly tractable form when M is a complex submanifold or a real hypersurface on which Ji is a principal vector for each unit normal ( to M. The rank of the focal map onto a sheet of the focal set of M is also computed in terms of the second fundamental form. In the case of a real hypersurface on which JE is principal with corresponding principal curvature ,u, if the map onto a sheet of the focal set corresponding to ,u has constant rank, then that sheet is a complex submanifold over which M is a tube of constant radius (Theorem 1). The other sheets of the focal set of such a hypersurface are given a real manifold structure in Theorem 2. These results are then employed as major tools in obtaining two classifications of real hypersurfaces in cPm. First, there are no totally umbilic real hypersurfaces in cPm, but we show: THEOREM 3. Let M be a connected real hypersurface in CPm, m > 3, with at most two distinct principal curvatures at each point. Then M is an open subset of a geodesic hypersphere. Secondly, we show that there are no Einstein real hypersurfaces in cPm and characterize the geodesic hyperspheres and two other classes of hypersurfaces in terms of a slightly less stringent requirement on the Ricci tensor in Theorem 4. One of the first results in the geometry of submanifolds is that an umbilic hypersurface M in Euclidean space must be an open subset of a hyperplane or sphere. The proof goes as follows: assume that the shape operator is a scalar multiple of the identity, A = AX, and use the Codazzi equation to show that X is constant. Then either X = 0, in which case M lies on a hyperplane, or the focal points fx(x) = x + (l/X)t, ( the unit normal, all coincide, and M lies on the sphere of radius 1 /X centered at the unique focal point. This simple idea suggests a plan of attack for classifying hypersurfaces in terms of the nature of the principal curvatures. Under fairly general conditions, the set of focal points corresponding to a principal curvature X can be given a differentiable Received by the editors December 19, 1980. Presented to the Society at its annual meeting in San Francisco, January 10, 1981. 1980 Mathematics Subject Classificatiom Primary 53B25, 53C40.

Real Hypersurfaces in Complex Space Forms

The study of real hypersurfaces in complex projective space CP n and complex hyperbolic space CH n began at approximately the same time as Munzner’s work on isoparametric hypersurfaces in spheres. A key early work was Takagi’s classification [669] in 1973 of homogeneous real hypersurfaces in CP n . These hypersurfaces necessarily have constant principal curvatures, and they serve as model spaces for many subsequent classification theorems. Later Montiel [501] provided a similar list of standard examples in complex hyperbolic space CH n . In this chapter, we describe these examples of Takagi and Montiel in detail, and later we prove many important classification results involving them. We also study Hopf hypersurfaces, focal sets, parallel hypersurfaces and tubes using both standard techniques of submanifold geometry and the method of Jacobi fields.

Isoparametric and Dupin Hypersurfaces

A hypersurface M n 1 in a real space-form R n , S n or H n is isoparametric if it has constant principal curvatures. For R n and H n , the classification of isoparametric hypersurfaces is complete and relatively simple, but as ´ Elie Cartan showed in a series of four papers in 1938-1940, the subject is much deeper and more complex for hypersurfaces in the sphere S n . A hypersurface M n 1 in a real space-form is proper Dupin if the number g of distinct principal curvatures is constant on M n 1 , and each principal curvature function is constant along each leaf of its corresponding principal foliation. This is an important genera- lization of the isoparametric property that has its roots in nineteenth century differential geometry and has been studied effectively in the context of Lie sphere geometry. This paper is a survey of the known results in these fields with emphasis on results that have been obtained in more recent years and discussion of important open problems in the field.

Geometry of hypersurfaces

Preface.- 1. Introduction.- 2. Submanifolds of Real Space Forms.- 3. Isoparametric Hypersurfaces.- 4. Submanifolds in Lie Sphere Geometry.- 5. Dupin Hypersurfaces.- 6. Real Hypersurfaces in Complex Space Forms.- 7. Complex Submanifolds of CPn and CHn.- 8. Hopf Hypersurfaces.- 9. Hypersurfaces in Quaternionic Space Forms.- Appendix A. Summary of Notation.- References.- Index.

Geometric applications of critical point theory to submanifolds of complex projective space

In a recent paper, [6], Nomizu and Rodriguez found a geometric characterization of umbilical submanifolds M n ⊂ R n+p in terms of the critical point behavior of a certain class of functions L p , p ⊂ R n+p , on M n . In that case, if p ⊂ R n+p , x ⊂ M n , then L p (x) = (d(x,p)) 2 , where d is the Euclidean distance function.

Distance functions and umbilic submanifolds of hyperbolic space

Here Lp is the Euclidean distance function, Lp(x) = \p — f(x)\\ where / is the immersion of M into E. Cecil [2] characterized the metric spheres in hyperbolic space H in terms of hyperbolic distance functions Lp as follows. THEOREM B. Let M, n > 2, be a connected, compact, differentiate manifold immersed in H. Every Morse function Lp has exactly two critical points if and only if M is embedded as a metric n-sphere in H.

Reducible Dupin submanifolds

Pinkalls standard constructions for obtaining a Dupin hypersurface W in ℝN from a Dupin hypersurface M in ℝn, N>n, are studied in the context of Lie sphere geometry. It is shown that a compact Dupin hypersurface W in ℝN with g distinct principal curvatures at each point is reducible to a compact Dupin hypersurface M in ℝn if and only if g=2.

Focal points and support functions in affine differential geometry

The notions of focal point and support function are considered for a nondegenerate hypersurfaceMn in affine spaceRn+1 equipped with an equiaffine transversal field. IfMn is locally strictly convex, these two concepts are related via an Index theorem concerning the critical points of the support functions onMn. This is used to obtain characterizations of spheres and ellipsoids in terms of the critical point behavior of certain classes of affine support functions.

AN AFFINE CHARACTERIZATION OF THE VERONESE SURFACE

AbstractIfM2 is a nondegenerate surface in a 4-dimensional Riemannian manifold