Jacques Lafontaine

Sur la complétude des variétés pseudo-riemanniennes

Abstract We discuss completeness for pseudo-riemannian manifolds, and give new examples of non-complete compact manifolds. The former are simply connected , the latter locally homogeneous .

Integration and Applications

If f is a function of a real variable with continuous derivative,

Manifolds: The Basics

\displaystyle{\int _{a}^{b}f^{{\prime}}(t)\,dt = f(b) - f(a).}

Analysis on Riemannian manifolds and Ricci curvature

We give a characterisation of Bieberbach manifolds which are geodesic boundaries of a compact flat manifold, and discuss the low dimensional cases, up to dimension 4.

Sur la geometrie systolique des varietes de Bieberbach

“The notion of a manifold is hard to define precisely.” This is the famous opening of Chapter III of Lecons sur la Geometrie des espaces de Riemann by Elie Cartan. It is followed by a stimulating heuristic discussion on the notion of manifold which can still be read with pleasure. For additional historic perspective we also mention Riemann’s inaugural lecture, translated with annotations for the modern reader in [Spivak 79].

A remark about static space times

The Gauss-Bonnet theorem is at the heart of the geometry of manifolds. It mixes topology (triangulations, cohomology spaces), differential geometry (index of singular points of vector fields) and Riemannian geometry. We do not have the space to illustrate all of these ideas in detail. To keep with the spirit of the book, the proofs we give will use differential geometry to the greatest extent possible. We nonetheless believe it would be interesting to sketch a purely Riemannian proof in this introduction. The price we pay is using certain notions that have not been introduced (geodesics, geodesic curvature), of which we give the idea.

The systolic constant of orientable Bieberbach 3-manifolds

In the preceding chapters we saw several ways to show that two open subsets of \(\mathbf{R}^{n}\), and more generally two manifolds, are not diffeomorphic.