On Tensor-Product Structures and Grassmannian Structures

As it was shown by several authors, the tangent bundle of a Grassmann manifold is a tensor product of two certain vector bundles. On the other hand, Th. Hangan studied a manifold with a structure on which the tangent bundle was isomorphic to the tensor product of two vector bundles. He called this structure a tensor-product structure. Th. Hangan's study was aimed mainly at flat tensor-product structures and the natural tensor-product structure on the Grassmann manifold. In this paper, some of his results for flat tensor-product structure are extended to general tensor-product structures. In §3, the notion of grassmannian structures, which is a extension of that of protective structures due to S. Kobayashi and T. Nagano [4] is defined. The natural correspondence between grassmannian structures and tensor-product structures are established. This correspondence leads us to the unique existence of a certain grassmannian structure for a give tensor-product structure. In this situation, we say that this grassmannian structure is determined by the given tensor-product structure. The notion of Cartan connection in a grassmannian structure is also introduced. Particularly, there exists uniquely so-called normal connection in a grassmannian structure determined by a tensor-product structure. Lastly, the local flatness of grassmannian structures is discussed. The consideration is made only for the real cases, but a similar discussion seems to be possible for the complex cases. The present author wishes to express his hearty thanks to Prof. Dr. M. Matsumoto for his kind encouragement. The author is also thankful to Prof. Dr. Y. Ichijyo who attracted my interests in this direction.

公開日:2010年1月24日で登録したコンテンツは、国立情報学研究所において電子化したものです。