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htaati
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how can I transform my calculation on S^3 to the S^2.
for example a trace or a Fourier transform
for example a trace or a Fourier transform
mathwonk said:hopf map? consider R^4 as C^2, the complex 2 space, and consider all complex "lines" in C^2. this family of complex subspaces is homeomorphic to S^2, hence this fibres S^3 over S^2 with fibers equal to circles. this is the famous hopf map from S^3 to S^2.
A manifold is a topological space that locally resembles Euclidean space. It can be thought of as a curved surface that can be smoothly mapped onto a flat plane.
A 2-dimensional manifold is a surface that can be embedded in 3-dimensional space, while a 3-dimensional manifold is a space that can be embedded in 4-dimensional space. Essentially, a 3-dimensional manifold has more "curvature" or higher dimensionality than a 2-dimensional manifold.
S^3 and S^2 refer to 3-dimensional and 2-dimensional spheres, respectively. Transforming calculations on S^3 to S^2 allows us to study the behavior of functions and their derivatives on curved surfaces, which has many applications in fields such as physics, engineering, and computer graphics.
Traditional calculus deals with functions on flat Euclidean spaces, while manifold calculus deals with functions on curved spaces. This means that the standard rules and formulas of calculus may not apply, and new techniques must be developed to handle the curvature and local structure of the manifold.
Examples of manifolds in the real world include the Earth's surface, which can be approximated as a 2-dimensional manifold, and the universe, which can be modeled as a 3-dimensional manifold. Other examples include the human brain, which has a complex 3-dimensional structure, and the space-time continuum, which is a 4-dimensional manifold.