What is “Riemannian Geometry”?
In: Mathematics
Differential geometry is a field of mathematics that applies the techniques developed in calculus and linear algebra to answer questions about geometry. Riemannian geometry is a branch of differential geometry dealing with certain “nice” mathematical spaces known as Riemannian manifolds. These spaces look very much like ordinary Euclidean space if you zoom in enough, and they are smooth and continuous spaces that have well-defined notions of angle, distance, area, volume, etc. so that you can talk about them using familiar geometric notions. However, at larger scales they may be curved in complex ways that makes them behave very differently than Euclidean space. Riemannian geometry has become a prominent branch of mathematics because it forms the main mathematical foundation for the general theory of relativity in physics.
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