Riemannian Geometry and Its Applications

Dear Colleagues,

Riemannian Geometry is a generalization of differential geometry. Differential geometry studies the geometry of curves and surfaces using Calculus and Linear Algebra. Riemannian Geometry studies smooth manifolds using a Riemannian metric. Locally, manifolds have properties of Euclidean spaces or other topological spaces, often in higher dimensions. Riemannian metrics express distances by means of smooth positive definite bilinear forms. Concepts in Euclidean geometry have natural analogues in Riemannian geometry. These include, but are not limited to, arc length of curves, areas of plane regions, volumes of solids and curvature.

There are many applications of Riemannian geometry to other branches of mathematics and to the sciences. Einstein used it and its generalization, Finsler geometry, to formulate general relativity theory. It impacted group theory, representation theory analysis, algebraic and differential topology and, more recently, statistics.

Information geometry applies the techniques of Riemannian geometry to probability and statistics. One regards probability distributions for a statistical model as the points of a Riemannian manifold. The Fisher information provides the Riemannian metric.

There are many diverse applications of information geometry. These include, for example, time series, machine learning, biology and mathematical finance.

Prof. Dr. Marvin H. J. Gruber
Guest Editor

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• affine connection
• alpha connection
• Christoffel symbols
• complete manifold
• curvature
• divergence
• Einstein’s equation
• exponential family
• flat manifold
• geodesics
• parallel translation
• parameters
• statistical manifold
• tangent space
• tensors