Compact real hypersurfaces in complex two-plane Grassmannians

Integral formulas in Riemannian geometry

Integral formulas in riemannian geometry

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.

/pdf/focal-sets-and-real-hypersurfaces-in-complex-projective-52an22teh6.pdf

Focal sets and real hypersurfaces in complex projective space

Let M be a real hypersurface in /"'(C), J be the complex structure and | denote a unit normal vector field on M. We show that M is (an open subset of) a homogeneous hypersurface if and only if M has constant principal curvatures and Jt, is principal. We also obtain a characterization of certain complex submanifolds in a complex projective space. Specifically, /""(C) (totally geodesic), Q", Pl(C) x P"(C). SU{5)/S(U{2) X (7(3)) and SO(10)/t/(5) are the only complex submani- folds whose principal curvatures are constant in the sense that they depend neither on the point of the submanifold nor on the normal vector.

/pdf/real-hypersurfaces-and-complex-submanifolds-in-complex-ymm1uahvhd.pdf

Real hypersurfaces and complex submanifolds in complex projective space

Given a real hypersurface of a complex hyperbolic space #x2102;?Hn,we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H12n+1.Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of ℂHn.

Compact real hypersurfaces in complex two-plane Grassmannians

Citations

Integral formulas in Riemannian geometry

References

Integral formulas in riemannian geometry

Integral formulas in Riemannian geometry

Focal sets and real hypersurfaces in complex projective space

Real hypersurfaces and complex submanifolds in complex projective space

On some real hypersurfaces of a complex hyperbolic space

Related Papers (5)

Real hypersurfaces in complex two-plane Grassmannians with parallel normal Jacobi operator

Linear spaces on rational hypersurfaces of odd degree

Rigidity theorems for closed hypersurfaces in space forms

Biharmonic hypersurfaces in complete Riemannian manifolds

Real analytic structures on a symplectic manifold