Finsler Manifolds with a Linear Connection

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In the previous paper [5], the present author has treated Finsler manifolds with such a property that the tangent spaces at arbitrary points are congruent (isometrically linearly isomorphic) to a single Minkowski space, and introduced, as a typical example of such a space, the notion of {V, H}-manifolds. At the same time, it has been shown that the {V, H}-manifolds are generalized Berwald spaces defined by Hashiguchi [3] and Wagner [9]. Now, the present paper has two main purposes. One is to consider the converse of the above-mentioned result. After some preparation, it will be proved, in §3, that if a generalized Berwald space is connected, it is actually a {V, H}-manifold. Next, in a Minkowski space is presented a Riemann metric, which is different from the Minkowski norm. Therefore, geodesic lines with respect to this Riemann metric can be introduced in the Minkowski space, which we call C-geodesics. A connected Finsler manifold M with a linear connection Γ^i_<jk>(x) is to be considered. With regard to arbitrary two points p and q in M and any piecewise differentiable curve C joining p and q, we can define a linear isomorphic mapping σ between the tangent Minkowski spaces Tp(M] and Tq(M) by parallel displacement with respect to Γ^i_<jk>(x) along the curve C. Now, the other main purpose of the present paper is to find a necessary and sufficient condition for a to map any C-geodesic in Tp(M) to a C-geodesic in Tq(M). It will be shown, in the last section, that the condition is C^i_<jk|h>=0 or equivalently P^i_<jkh>=0 with respect to the Finser connection associated with the linear connection Γ^i_<jk>(x).
1976-10-31

Finsler Manifolds with a Linear Connection

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