On Rund's connection


We show that the holomorphic curvature K<sub>F</sub> (associated with a complex Finsler metric F)in the sense of M. Suzuki, [13] ,and B. Wong,[15], is (in the smooth case) precisely the holomorphic curvature of a connection essentially due to H. Rund, [12] (and reproposed in the bundle-theoretic setting by S. Kobayashi, [8]). We prove a complex analogue of Deike's theorem in real Finsler geometry. The indicatrix in each fibre of a convex complex Finsler bundle is shown to be an extrinsic sphere.

DOI Code: 10.1285/i15900932v15n1p85

Classification: 51H10; 51H20; 51A35

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