Riemannian geometry notes: It is a branch of differential geometry that focuses on the study of smooth manifolds equipped with Riemannian metrics. It is named after the 19th-century German mathematician Bernhard Riemann. Riemannian geometry has important applications in various areas of mathematics and physics, particularly in the theory of relativity and differential geometry.

##### Some important topics in Riemannian geometry:

**Manifolds:**In Riemannian geometry, the primary objects of study are smooth manifolds.**Riemannian Metrics:**A Riemannian metric is a smoothly varying inner product structure defined on the tangent space at each point of a manifold.**Geodesics:**Geodesics are curves on Riemannian manifolds that generalize straight lines in Euclidean space. They are paths of shortest distance between two points on the manifold, as measured by the Riemannian metric. Geodesics play a crucial role in understanding the geometry of the manifold.**Curvature:**Curvature measures how a Riemannian manifold deviates from being flat (like Euclidean space).**Connection:**A connection is a mathematical structure that defines how to differentiate vector fields on a manifold.**Covariant Differentiation:**The Levi-Civita connection also leads to the concept of covariant differentiation, which is a generalization of ordinary differentiation that accounts for the curvature of the manifold.**Applications:**Riemannian geometry has a wide range of applications in mathematics and physics.

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Riemannian Geometry notes