In this book, the reader can find an elegant approach to some relevant ordinary and partial differential equations within the language of local Riemannian Geometry. In fact, the subtitle of the book, namely, “Applications to Differential Equations” can be seen as the core of the volume, so that the construction of the underlying geometry of Riemann spaces is concentrated to the study of these equations. The methodology is direct and efficient: the presentations of examples and properties along the book provides the motivation for the subsequent introduction of the tensors and the Riemannian definitions that, afterwards, are applied back to the equations. This serves to get useful techniques for the computation of solutions, the determination of conservations laws, the explanation of the transformations of the systems of equations and, finally, the symmetries that these systems possess. The main source of models come, specially in the last chapters of the book, from General Relativity.

The reader looking for a reference on standard pure Riemannian Geometry solely, will be perhaps disappointed. For example, the book seems not even contain the word “manifold”. The presentations of the Riemannian ideas and properties are always given in local coordinates as they are aimed at their applications to the differential equations. Global or topological questions are not tackled. As we have explained above, the goal of the book is not that. The author, an expert on geometric ODEs and PDEs and with a long experience in the teaching of the subject, has focus the attention of the book on the geometric study and applications of differential equations. This provides a common framework to apparently different situations that, at the end, become elegantly unified. Geometers as well as students and researchers on differential equations, with basic notions on differential geometry, will find this book of great interest.

Reviewer:

Marco Castrillon Lopez