Almost Complex and Complex Structures

The following sections are included:

• Dedication

• Contents

• Preface

This chapter contains some fundamental definitions, formulas and theorems for Riemannian manifolds for the later uses in this books.

In this chapter we study the almost complex structures of a differentiable manifold. We give a definition in Sec. 1 and an example for the unit 6-sphere S⁶ in Sec. 2. Section 3 contains a discussion of the complexification of a vector space over the set of all real numbers. In Sec. 4 we present some necessary and sufficient conditions for a differentiable 2n-manifold M²ⁿ to admit an almost complex structure J_i^j. In Sec. 5 some identities for the relationship between J_i^j and the Riemannian curvature tensor R_hijk with respect to a Riemannian metric g_ij of Men are used to define some classes of almost complex structures on M²ⁿ, and the inclusion relations among those classes are proved.

In this chapter we use covariant derivatives of a general almost Hermitian structure with respect to its Hermitian metric to define the most known special almost Hermitian structures, and give some properties as well as some inclusion relations of those structures.

In this chapter we shall study the commutative relationship among the real Laplacian Δ and some real operators on an almost Hermitian manifold, and the spectral geometry of Riemannian and almost Hermitian manifolds.

In this chapter we shall study various complex operators including complex Laplacians on Riemannian manifolds with an almost complex structure.

The notion of a complex manifold is a natural outgrowth of that of a differentiable manifold. Its importance lies to a large extent in the fact that the complex manifolds include the complex algebraic varieties and the Riemannian surfaces as special cases, and furnish the geometric basis for functions of several complex variables. Its development has led to clarifications of classical algebraic geometry and to new results and problems. A fundamental problem in the theory of complex manifolds is to characterize the orientable manifolds of even dimensions 2n which can be given a complex structure.

By using characteristic classes and cohomology operations on the classes, in particular, the Steenrod squaring and reduced power operations, various necessary conditions are known for a manifold to be almost complex and therefore to be complex since a complex structure is naturally an almost complex structure. These conditions suffice to show that among the even-dimensional spheres only the 2-sphere S² and the 6-sphere S⁶ are almost complex. In fact, the absence of an almost complex structure on S^4k for k≥1 and S²ⁿ for n ≥ 4 was proved by Wu [174] and jointly by Borel and Serre [18] respectively. Kirchhoff [107], [108] has shown that if Sⁿ admits an almost complex structure, then Sⁿ⁺¹ admits an absolute parallelism, and Adams [2] has shown that Sⁿ⁺¹ a dmits an absolute parallelism only for n + 1 = 1, 3 and 7. The result of Adams combined with that of Kirchhoff implies the result of Wu, Borel and Serre.

On the other hand, for a given almost complex structure, necessary conditions are known in order that it defines a complex structure (Ehresmann and Libermann [50], Eckmann and Frohfisher [47]). These conditions are also sufficient, if the almost complex structure is analytic.

In 1986 the present author [97] of this book solved the long standing unsolved problem: Does there exist a complex structure on S⁶? The purpose of this chapter is mainly to present the solution in details.

This chapter deals with complex structures. A definition and several examples are given in Secs. 1 and 2 respectively. In Sec. 3 relations to almost complex structures are discussed. Section 4 contains integrability conditions for an almost complex structure. In Sec. 5, we present the calculations, in some details, of the operator ∂ for 0-, 1- and 2-forms respectively. In Sec. 6 we derive some necessary conditions for an almost complex structure to be a complex structure, and we use those conditions to prove the nonexistence of a complex structure on the 2n-sphere S²ⁿ for n ≥ 2.

This chapter contains some vanishing theorems for Riemannian and Kählerian manifolds; by vanishing theorems here we mean those giving conditions for the manifolds to have some vanishing homology groups or Killing tensor fields. We shall use both the maximum modular principle method and the integral formula method.

The following sections are included:

• Bibliography

• Index