Applications of Bott connection to Finsler geometry (Q2733924)

In this paper the connection theory of a complex Finsler space is investigated by the complex Bott connection \(D\) which is a partial connection defined by a splitting of the fundamental sequence NEWLINE\[NEWLINE0\to T_{E/M} @>i >> T_E@>{d\pi}>> \pi^*T_M\to 0,NEWLINE\]NEWLINE where \(\pi:E\to M\) is a holomorphic vector bundle over a complex manifold, and \(T_{E/M}\) denotes the relative tangent bundle of \(\pi\). First the Bott connection is defined and the coefficients of its curvature are investigated . It is proved that if the Bott of connection \(D\) of a convex vector bundle \((E,\|\cdot\|)\) is flat, then: a) the fundamental sequence splits holomorphically, and b) there exists a flat Hermitian metric on \(E\) (\(\|\cdot \|\) means the Finsler norm). Then the Bott connection \(D\) of \((E,\|\cdot \|)\) is extended to an ordinary connection \(\nabla\) on \(T_{E/M}\). NEWLINENEWLINENEWLINEIt is proved that \((E,\|\cdot\|)\) is flat iff: a) \(R^\nabla=0\) or b) it is modeled on a complex Minkowski space and its associated Hermitian bundle \((E,h_F)\) is flat [see \textit{T. Aikou}, Publ. Math. 54, No. 1-2, 165-179 (1999; Zbl 0926.53013)]. Finally: a holomorphic vector bundle admits a convex Finsler metric iff the projective bundle \(\pi_{P_E}:P_E\to M\) associated with \(E\) is a Kähler morphism.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00031].