Riemannian Geometry
Introduction
Riemannian geometry is a branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e., with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume.
History
The field of Riemannian geometry originated in the work of the German mathematician Bernhard Riemann, who defined the concept of a Riemannian manifold and an intrinsic curvature for such manifolds. His ideas were then formalized and further developed by other mathematicians such as Elie Cartan and Hermann Weyl.
Basic Concepts
Riemannian Manifold
A Riemannian manifold is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p ↦ g_p(X(p),Y(p)) is a smooth function.
Riemannian Metric
The Riemannian metric g is a smoothly varying, symmetric, positive-definite bilinear form on each tangent space. This gives a notion of length to tangent vectors, and, by integration, a notion of length to curves.
Geodesics
Geodesics are locally distance-minimizing curves in a Riemannian manifold. They are the generalization of the concept of a "straight line" to "curved spaces".
Curvature
The curvature of a Riemannian manifold is a measure of how much the geometry of the manifold deviates from that of flat space. It is defined in terms of the Riemann curvature tensor.
Advanced Topics
Connections and Covariant Derivatives
In Riemannian geometry, a Levi-Civita connection is used to define the covariant derivative, a way of taking derivatives of vector fields that respects the Riemannian metric.
Riemannian Submanifolds
Riemannian submanifolds are submanifolds of a Riemannian manifold that inherit a Riemannian metric from the ambient manifold.
Curvature Tensor
The Riemann curvature tensor is a geometric object which measures the extent to which the geometry of a Riemannian manifold deviates from being flat.
Ricci Curvature
The Ricci curvature is a measure of the extent to which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space.
Scalar Curvature
The scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point.
Applications
Riemannian geometry has found applications in various fields of science and engineering. For instance, it is used in physics, particularly in the theory of general relativity, where the structure of spacetime is modeled as a four-dimensional, Lorentzian manifold. It is also used in computer science, in the field of computer vision for shape analysis, and in the analysis of high-dimensional data.