Transformations of Manifolds and Applications to Differential Equations

The interaction between differential geometry and differential equations is the general theme of this analysis. The author focuses on the role played by differential geometry in the study of differential equations, combining the geometric and analytic aspects of the theory, not only in the classical examples, but also in results obtained since 1980, on integrable systems with an arbitrary number of independent variables. The discussion applies to problems in evolution equations, strongly hyperbolic systems of the hydrodynamic type, linear Weingarten surfaces, and submanifolds of constant curvature.