Geodesic monotone vector fields
Geodesic convex functions on open geodesic convex subsets of Riemannian manifolds have been considered by \textit{C. Udriste} [Studii Cerc. Mat. 28, 735-745 (1976; Zbl 0372.53017); Bull. Math. Soc. Sci. Math. Repub. Soc. Roum., Nouv. Ser. 21(69), 215-218 (1977; Zbl 0372.53018)] and by \textit{T. Rapcsak} [Smooth nonlinear optimization of \(R^n\). Nonconvex Optimization and Its Applications. 19. Dordrecht: Kluwer Academic Publishers. xiii, 374 p. (1997)]. The author investigates geodesic monotone vector fields. He proves that a vector field \(X\) on a Riemannian manifold is geodesic monotone (strictly geodesic monotone) if and only if for all deformations of any geodesic arc \(\gamma\) inducing the infinitesimal variation equal to the restriction of \(X\) to \(\gamma\), the first variation of the length of \(\gamma\) is nonnegative (positive). On Riemannian manifolds with a closed geodesic or on compact Riemannian manifolds there is no strictly monotone vector field. If every geodesic of a Riemannian manifold \(M\) is closed there is no virtually monotone vector field on \(M\). The author defines on a complete simply connected Riemannian manifold of non positive sectional curvature for a smooth strictly increasing function \(f\) the so called \(f\)-position vector field and shows that such a vector field \(X\) is strictly geodesic monotone. For a vector field \(X\) on a Riemannian manifold the author defines the lower scalar geodesic derivative \(\underline{X}^\#\) (the upper geodesic scalar derivative \(\overline{X}_\#\)) and discusses the connections between the geodesic monotonicity of \(X\) and \(\underline{X}^\#\).
scientific article; zbMATH DE number 1463549