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On a notable connection in Finsler geometry. (English) Zbl 0787.53018
In the \(n\)-dimensional manifold \(M\) the metric function \(F(x,y)\) (homogeneous of degree one in \(y\)) is given. The metric tensor is obtained from \(F\) in the usual manner. In the vector bundle \(\pi^* TM\) the local orthonormal basis \(\{e_ i\}\) is introduced, where \(e_ j = u_ j^ l\;\partial/\partial x^ l\), \(e_ n = y_ i/F\;\partial/\partial x^ i\). The dual co-frame field is \(\{w^ j\}\), where \(w^ i = v^ i_ kdx^ k\), \(w^ n = F_{y^ k}dx^ k\). The connection \(\nabla\) is defined by \(\nabla_{e_ k} = w^ i_ k\otimes e_ i\), \(\nabla \partial /\partial x^ k = \Gamma^ i_ k\otimes \partial/\partial x^ i\), \(\Gamma^ i_ k = \Gamma^ i_{kl}dx^ l\), where \(\Gamma_{jkl}\) are the Berwald connection coefficients. Using this connection the geodesics and Jacobi fields are examined. The second variation of arc length in terms of the curvature tensor is obtained.
Reviewer: I.Comic (Novi Sad)

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C22 Geodesics in global differential geometry