Almost Complex Projective Connections

概要

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The theory of real projective connections standing on the view-point of the theory of vector bundles was investigated perfectly by T. Otsuki. Its extension to generalized spaces was done by the present author and K. Eguchi. On the other hand, T. Ishihara studied complex projective structures and found the close relation between the complex projective structures and H-projective connections which had been introduced by T. Tashiro and developed by S. Ishihara and many other authors. In the present paper, our main purpose is to study manifolds endowed with almost complex projective structures. First of all, in §§1 and 2, we consider an almost complex projective vector bundle introduced in [4] and several distributions which define an almost complex projective connection. This connection, however, can be determined also by an usual differential geometric method. In §3, we find a certain distribution P which is intrinsic to the almost complex projective connection. A mapping σ which gives a correspondence between any two fibers in the almost complex projective vector bundle is treated in §4. And we show that the mapping σ preserves the complex projective structure in each fiber invariant if the distribution P is integrable. The last section is devoted to study a manifold where the given distribution P is integrable, for example, we obtain that if P is integrable then the manifold is an H-projectively flat complex manifold.