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.

Real Hypersurfaces in Complex Space Forms

It is known that there do not exist real hypersurfaces with parallel curvature tensor in quaternionic projective spaces. In this paper we classify real hypersurfaces of quaternionic projective space whose curvature tensor is parallel in the direction of certain 3-dimensional distribution.

Real hypersurfaces of quaternionic projective space satisfying ▽UiR = 0

We classify real hypersurfaces in complex projective spaces whose structure Jacobi operator is Lie parallel in the direction of the structure vector field.

Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie ξ-parallel

In this paper we consider a new notion of D ⊥ -parallel shape operator for real hypersurfaces in complex two-plane Grassmannians G2(C m+2 ) and give a non-existence theorem for a Hopf hypersurface in G2(C m+2 )w ithD ⊥ -parallel shape operator. Mathematics Subject Classification (2010). Primary 53C40; Secondary 53C15.

Real Hypersurfaces in Complex Two-plane Grassmannians with D ⊥ -Parallel Shape Operator

Canonical Centroaffine Hypersurfaces in R n+1

We prove the nonexistence of real hypersurfaces in nonflat complex space forms whose Jacobi operator associated to the structure vector field is parallel. In order to prove this result we also obtain the nonexistence of several classes of non homogeneous real hypersurfaces in complex projective space.

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Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in Nonflat Complex Space Forms

We prove the non existence of real hypersurfaces in complex projective space whose structure Jacobi operator is parallel in any direction of the maximal holomorphic distribution.

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Real hypersurfaces in complex projective space whose structure Jacobi operator is $\mathbb D$-parallel

We classify real hypersurfaces of complex projective space C P m , m ⩾ 3 , with D -recurrent structure Jacobi operator and apply this result to prove the non-existence of such hypersurfaces with recurrent structure Jacobi operator.

Florentino G. Santos

Papers

Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie ξ-parallel

Non-Existence of Real Hypersurfaces with Parallel Structure Jacobi Operator in Nonflat Complex Space Forms

Real hypersurfaces in complex projective space whose structure Jacobi operator is $\mathbb D$-parallel

Real hypersurfaces in complex projective space with recurrent structure Jacobi operator

On the Pick Invariant, the Affine mean Curvature and the Gauss Curvature of Affine Surfaces