Fuchsian reduction is a method for representing solutions of nonlinear PDEs near singularities. The technique has multiple applications including soliton theory, Einstein's equations and cosmology, stellar models, laser collapse, conformal geometry and combustion. Developed in the 1990s for semilinear wave equations, Fuchsian reduction research has grown in response to those problems in pure and applied mathematics where numerical computations fail.

The main subject of this book is an up-to-date and in-depth survey of the theory of normal frames and coordinates in di?erential geometry. The existing results, as well as new ones obtained lately by the author, on the theme are presented. The text is so organized that it can serve equally well as a reference manual, introduction to and review of the current research on the topic. Correspondingly, the possible audience ranges from graduate and post-graduate students to sci- tists working in di?erential geometry and theoretical/mathematical physics. This is re?ected in the bibliography which consists mainly of standard (text)books and journal articles. The present monograph is the ?rst attempt for collecting the known facts concerting normal frames and coordinates into a single publication. For that r- son, the considerations and most of the proofs are given in details. Conventionally local coordinates or frames, which can be holonomic or not, are called normal if in them the coe?cients of a linear connection vanish on some subset, usually a submanifold, of a di?erentiable manifold. Until recently the ex- tence of normal frames was known (proved) only for symmetric linear connections on submanifolds of a manifold. Now the problems concerning normal frames for derivationsof thetensor algebraovera di?erentiablemanifoldarewellinvestigate; in particular they completely cover the exploration of normal frames for arbitrary linear connections on a manifold. These rigorous results are important in conn- tion with some physical applications.

In this second edition, the main additions are a section devoted to surfaces with constant negative curvature, and an introduction to conformal geometry. Also, we present a -soft-proof of the Paul Levy-Gromov isoperimetric inequal­ ity, kindly communicated by G. Besson. Several people helped us to find bugs in the. first edition. They are not responsible for the persisting ones! Among them, we particularly thank Pierre Arnoux and Stefano Marchiafava. We are also indebted to Marc Troyanov for valuable comments and sugges­ tions. INTRODUCTION This book is an outgrowth of graduate lectures given by two of us in Paris. We assume that the reader has already heard a little about differential manifolds. At some very precise points, we also use the basic vocabulary of representation theory, or some elementary notions about homotopy. Now and then, some remarks and comments use more elaborate theories. Such passages are inserted between *. In most textbooks about Riemannian geometry, the starting point is the local theory of embedded surfaces. Here we begin directly with the so-called "abstract" manifolds. To illustrate our point of view, a series of examples is developed each time a new definition or theorem occurs. Thus, the reader will meet a detailed recurrent study of spheres, tori, real and complex projective spaces, and compact Lie groups equipped with bi-invariant metrics. Notice that all these examples, although very common, are not so easy to realize (except the first) as Riemannian submanifolds of Euclidean spaces.

Metric theory has undergone a dramatic phase transition in the last decades when its focus moved from the foundations of real analysis to Riemannian geometry and algebraic topology, to the theory of infinite groups and probability theory.

Radiolaria are a very diverse marine siliceous microplankton group that have existed at least snice the Cambrian to the recent. This volume gives a representative view of research topics discussed at the 10th International Meeting of Radiolarian Palaeontologists. The articles of this volume cover mainly radiolarian biochronology and radiolarian fauna changes.