The object of this book is to present the basic facts of convex
functions, standard dynamical systems, descent numerical algorithms and
some computer programs on Riemannian manifolds in a form suitable for
applied mathematicians, scientists and engineers. It contains
mathematical information on these subjects and applications distributed
in seven chapters whose topics are close to my own areas of research:
Metric properties of Riemannian manifolds, First and second variations
of the p-energy of a curve; Convex functions on Riemannian manifolds;
Geometric examples of convex functions; Flows, convexity and energies;
Semidefinite Hessians and applications; Minimization of functions on
Riemannian manifolds. All the numerical algorithms, computer programs
and the appendices (Riemannian convexity of functions f: R R, Descent
methods on the Poincare plane, Descent methods on the sphere,
Completeness and convexity on Finsler manifolds) constitute an attempt
to make accesible to all users of this book some basic computational
techniques and implementation of geometric structures. To further aid
the readers, this book also contains a part of the folklore about
Riemannian geometry, convex functions and dynamical systems because it
is unfortunately "nowhere" to be found in the same context; existing
textbooks on convex functions on Euclidean spaces or on dynamical
systems do not mention what happens in Riemannian geometry, while the
papers dealing with Riemannian manifolds usually avoid discussing
elementary facts. Usually a convex function on a Riemannian manifold is
a real- valued function whose restriction to every geodesic arc is
convex.
