Euclidean Space - Generalizations

Generalizations

In modern mathematics, Euclidean spaces form the prototypes for other, more complicated geometric objects. For example, a smooth manifold is a Hausdorff topological space that is locally diffeomorphic to Euclidean space. Diffeomorphism does not respect distance and angle, so these key concepts of Euclidean geometry are lost on a smooth manifold. However, if one additionally prescribes a smoothly varying inner product on the manifold's tangent spaces, then the result is what is called a Riemannian manifold. Put differently, a Riemannian manifold is a space constructed by deforming and patching together Euclidean spaces. Such a space enjoys notions of distance and angle, but they behave in a curved, non-Euclidean manner. The simplest Riemannian manifold, consisting of Rn with a constant inner product, is essentially identical to Euclidean n-space itself.

If one alters a Euclidean space so that its inner product becomes negative in one or more directions, then the result is a pseudo-Euclidean space. Smooth manifolds built from such spaces are called pseudo-Riemannian manifolds. Perhaps their most famous application is the theory of relativity, where empty spacetime with no matter is represented by the flat pseudo-Euclidean space called Minkowski space, spacetimes with matter in them form other pseudo-Riemannian manifolds, and gravity corresponds to the curvature of such a manifold.

Our universe, being subject to relativity, is not Euclidean. This becomes significant in theoretical considerations of astronomy and cosmology, and also in some practical problems such as global positioning and airplane navigation. Nonetheless, a Euclidean model of the universe can still be used to solve many other practical problems with sufficient precision.