COMPLETENESS OF FINSLER MANIFOLDS

This paper analyses some constructions that produce complete Finsler manifolds: 1) Let the Finsler manifolds (M, g(x, y)) and (M,gBAR(x, y)) be given. Then (M,gBAR(x, y)) is complete if (M, g(x, y)) is complete and the tensor field gBAR - g is positive semi-definite. 2) If (M,g(x,y)) is a Finsler manifold and f : M --> R is a proper function then the Finsler manifold (M, g(x,y)+df(x)xdf(x)) is complete. Using this construction we prove that a Finsler manifold which supports a proper function whose differential has bounded relative length is complete. 3) Let the Finsler manifolds (M1, g1 (x1, y1)) and (M2 , g2 (x2, y2)) be given and suppose that f > 0 is a differentiable function on M1. The warped product (M1 x M2, g1 + fg2) is complete if aiid only if (M1, g1 (x1, y1)) and (M2,g2(x2, y2)) are complete.