Elements of Differential Geometry

Abstract

Physics deals with the behavior of certain objects (particles, bodies, fields, etc.) in 3-space in course of time. Formally, time and space can be combined to a 4-dimensional space-time, though the two entities, space and time, are always clearly distinguished. Usually 4-dimensional space-time is described by the mathematical picture of a manifold, a 4-dimensional space that locally looks like the Euclidean $\mathbb {R}^4$. Clearly this manifold picture is an idealization since arbitrarily small distances in space or time cannot be measured in principle. Moreover, because of quantum effects, one expects the manifold picture to break down for distances of order the Planck-length l _P = (Għ∕c ³)^1∕2 ≃ 1, 6 × 10⁻³⁵ m or smaller. Presently, it is unclear how to describe gravitational physics at such small length scales.