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hansenscane
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Does anyone know what the Asian characters on this book mean? Why are they there?
A Riemannian manifold is a mathematical concept that combines the ideas of a manifold, which is a topological space that locally resembles Euclidean space, and a Riemannian metric, which is a way of measuring distances and angles on the manifold. Essentially, it is a smooth, curved space with a consistent way of measuring distances and angles at each point.
Riemannian manifolds are important in many areas of mathematics and physics, including differential geometry, topology, and general relativity. They provide a framework for studying smooth, curved spaces and have applications in fields such as computer graphics, robotics, and machine learning.
John M. Lee is a mathematician and professor at the University of Washington. He is a leading expert in the field of Riemannian geometry and has written several books, including "Riemannian Manifolds: An Introduction to Curvature" which is a widely used textbook in the subject.
A Euclidean space is a flat, infinite, and homogeneous space, while a Riemannian manifold is a curved, finite, and possibly non-homogeneous space. In a Riemannian manifold, the distance between two points and the angle between two curves may vary depending on the point of measurement, whereas in a Euclidean space, these measurements are constant.
Riemannian manifolds are essential in the theory of general relativity, which describes the curvature of spacetime caused by the presence of matter and energy. In this theory, the universe is modeled as a four-dimensional Riemannian manifold, and the curvature of this manifold is related to the gravitational force. Riemannian manifolds are also used in other areas of physics, such as the study of black holes and cosmology.