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S.Z. Nemeth
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Geodesic monotone vector fields

*(Lobachevskii Journal of Mathematics,
Vol.5, pp.13-28)*

Having in mind the Minty-Browder monotonicity notion, we shall
generalize it for vector fields on Riemannian manifolds, defining the
geodesic monotone vector fields. The gradients of geodesic convex functions,
important in optimization, linear and nonlinear programming on Riemannian
manifolds, are geodesic monotone vector fields.
The geodesic monotonicity will be related with the first variation of
the length of a geodesic.
The connection between the existence of closed geodesics and monotone vector
fields will also be analyzed.
We give a class of strictly monotone vector fields on a
simply connected, complete Riemannian manifold with nonpositive sectional
curvature, which generalize the notion of position vector fields.
The notion of geodesic scalar derivative
will be introduced for characterization of geodesic monotone vector
fields on such manifolds.
The constant sectional curvature case will be analized separately, since it
has important peculiarities.

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