one of the world's leading differential geometers and a corresponding member of the French Academy of Sciences for half a century, passed away on October 15, 2016, at the age of eighty-nine.Marcel Berger's contributions to geometry were both broad and deep.The classification of Riemannian holonomy groups provided by his thesis has had a lasting impact on areas ranging from theoretical physics to algebraic geometry.His 1960 proof that a complete oriented even-dimensional manifold with strictly quarter-pinched positive curvature must be a topological sphere is the

We study the free boundary Euler equations with surface tension in three spatial dimensions, showing that the equations are well-posed if the coefficient of surface tension is positive. Then we prove that under natural assumptions, the solutions of the free boundary motion converge to solutions of the Euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity.

Given an odd-dimensional compact manifold and a contact form, we consider the group of contact transformations of the manifold (contactomorphisms) and the subgroup of those transformations that precisely preserve the contact form (quantomorphisms). If the manifold also has a Riemannian metric, we can consider the L^2 inner product of vector fields on it, which by restriction gives an inner product on the tangent space at the identity of each of the groups that we consider. We then obtain right-invariant metrics on both the contactomorphism and quantomorphism groups. We show that the contactomorphism group has geodesics at least for short time and that the quantomorphism group is a totally geodesic subgroup of it. Furthermore we show that the geodesics in this smaller group exist globally. Our methodology is to use the right invariance to derive an “Euler–Arnold” equation from the geodesic equation and to show using ODE methods that it has solutions which depend smoothly on the initial conditions. For global existence we then derive a “quasi-Lipschitz” estimate on the stream function, which leads to a Beale–Kato–Majda criterion which is automatically satisfied for quantomorphisms. Special cases of these Euler–Arnold equations are the Camassa–Holm equation (when the manifold is one-dimensional) and the quasi-geostrophic equation in geophysics.

Following Ebin and Marsden (Ann Math 92(1):102–163, 1970) we provide a concise proof of the well-posedness of the equations of perfect fluid motion. We use a construction which casts the equations as an ordinary differential equation on a non-linear function space.

We study the free boundary Euler equations in two spatial dimensions. We prove that if the boundary has constant curvature, then solutions of the free boundary fluid motion converge to solutions of the Euler equations in a fixed domain when the coefficient of surface tension tends to infinity.

Let M be a compact manifold with a symplectic form ω and consider the group \({\mathcal{D}_\omega}\) consisting of diffeomorphisms that preserve ω. We introduce a Riemannian metric on M which is compatible with ω and use it to define an L 2-inner product on vector fields on M. Extending by right invariance we get a weak Riemannian metric on \({\mathcal{D}_\omega}\) . We show that this metric has geodesics which come from integral curves of a smooth vector field on the tangent bundle of \({\mathcal{D}_\omega}\) . Then, estimating the growth of such geodesics, we show that they extend globally.

Basic concepts and results Toponogov's theorem Homogeneous spaces Morse theory Closed geodesics and the cut locus The sphere theorem and its generalizations The differentiable sphere theorem Complete manifolds of nonnegative curvature Compact manifolds of nonpositive curvature Bibliography Additional bibliography Index.

The equations of the dynamics of an elastic material are a non- linear hyperbolic system whose unknowns are functions of space and time. If the material is incompressible, the system has an additional pseudo-dierential term. We prove that such a system has global (classical) solutions if the initial data are small. This contrasts with the case of compressible materials for which F. John has shown that such solutions may not exist even for arbitrarily small data.

I prove that the initial-value problem for the motion of a certain type of elastic body has a solution for all time if the initial data are sufficiently small. The body must fill all of three-space, obey a "neo-Hookean" stress-strain law, and be incompressible. The proof takes advantage of the delayed singularity formation which occurs for solutions of quasilinear hyperbolic equations in more than one space dimension. It turns out that the curl of the displacement of the body obeys such an equation. Thus, using Klainerman's inequality, one derives the necessary estimates to guarantee that solutions persist for all time.