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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.