The study of Bott connection and its applications to Finsler geometry

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In this research, we have investigated complex Bott connections and its applications to complex Finsler geometry in the period 2001-2002. In 2001, we have mainly studied some relations between the complex Bott connections and complex Finsler connections. Under the results obtained in 2001, we have studied the flatness and projective flatness of complex Finsler metrics, and moreover, the differential geometry of Kahler fibrations with pseudo Kahler metrics in 2002.The main contents of this research is the investigation of complex Finsler connection which is introduced on the relative tangent bundle over the projective bundle. In terms of this connection, we can investigate the projective flatness of complex Finsler metrics, and finally we obtained the projective curvature which is the obstruction of projective flatness. Such an investigation leads us naturally to the study of minimal ruled surface over a compact Riemann surface. In fact, the Kahler metric on such a surface induces a complex Finsler metric with negative curvature in the corresponding vector bundle. The basic and important fact is that the projective flatness of the corresponding Finsler metric is equivalent to that the minimal ruled surface is the total space of a Kahler submersion with isometric fibers to the base Riemann surface.The main results in this research are contained in "Kahler fibrations and complex Finsler geometry (preprint, 2002)" and "A note on some special Finsler manifold (preprint, 2002)".