Jean-Marie Souriau

Physics and geometry

Differential geometry, the contemporary heir of the infinitesimal calculus of the 17th century, appears today as the most appropriate language for the description of physical reality. This holds at every level: The concept of “connexion,” for instance, is used in the construction of models of the universe as well as in the description of the interior of the proton. Nothing is apparently more contrary to the wisdom of physicists; all the same, “it works.” The pages that follow show the conceptual role played by this geometry in some examples—without entering into technical details. In order to achieve this, we shall often have to abandon the complete mathematical rigor and even full definitions; however, we shall be able to give a precise description of the connection of ideas thanks to some elements of group theory.

Anisotropic Hubble expansion of large scale structures

With the aim of understanding the cosmic velocity fields at large scale, we investigate the dynamics of a pressureless distribution of gravitational sources moving under an anisotropic generalization of Hubble expansion and according to Euler-Poisson equations system. As a result, it turns out that such a behavior requires the distribution to be homogenous, similarly to Hubble law. Among several solutions, we show a planar kinematics which admits a constant (eternal) and rotational distortion, where the velocity field is not potential. Within this class, the one with no rotational distortion identifies to a bulk flow. To apply this model within cosmic structures as the Local Super Cluster, the solutions are interpreted as approximations providing us with an hint on the behavior of the cosmic flow just after decoupling era up to present date. Such a result suggests that the observed bulk flow may not be due exclusively to tidal forces but has a primordial origin.

Measures on a manifold

In this chapter we are going to extend the notion of a manifold by using the following conventions.