pszeudoRiemann

Pseudo-Riemann geometry is a mathematical concept that generalizes Riemannian geometry by relaxing the positive-definiteness of the metric tensor. Introduced by mathematician Mikhail Gromov in the 1990s, this extension allows for the study of geometric structures on manifolds with possibly degenerate metrics.

In Riemannian geometry, a manifold is equipped with a Riemannian metric, a smooth, positive-definite inner product on the tangent space at each point. This positive-definiteness ensures that the metric induces a smoothly varying notion of length and angle on the manifold. In contrast, a pseudo-Riemannian manifold is equipped with a degenerate metric, meaning that the inner product on the tangent space at some points is not positive-definite.

Pseudo-Riemannian geometry has applications in several areas of theoretical physics, including general relativity and supergravity. In these theories, the metric tensor can be degenerate, and pseudo-Riemannian geometry provides a mathematical framework for analyzing the curvature and properties of the spacetime manifold. The geometry of pseudo-Riemannian manifolds is also relevant to the study of singularities in curved spacetimes and the behavior of matter and energy under extreme conditions.

Despite its relatively recent introduction, pseudo-Riemannian geometry has already found its way into various areas of mathematics and physics, and continues to be an active area of research. The study of pseudo-Riemannian manifolds has led to insights into the nature of spacetime, and has opened up new avenues of investigation in differential geometry and theoretical physics. However, the mathematical framework of pseudo-Riemannian geometry remains an active area of research, with ongoing efforts to deepen our understanding of these structures and their properties.