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In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (a tangent plane) to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer ...
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. Begin with a sphere centered on the origin. Every line through the origin pierces the sphere in two opposite points called antipodes .
The study of maps of 1-dimensional manifolds are a non-trivial area. For example: Groups of diffeomorphisms of 1-manifolds are quite difficult to understand finely [2] Maps from the circle into the 3-sphere (or more generally any 3-dimensional manifold) are studied as part of knot theory.
Familiar examples of two-dimensional manifolds include the sphere, torus, and Klein bottle; this book concentrates on three-dimensional manifolds, and on two-dimensional surfaces within them. A particular focus is a Heegaard splitting, a two-dimensional surface that partitions a 3-manifold into two handlebodies. It aims to present the main ...
Riemannian manifolds were first conceptualized by their namesake, German mathematician Bernhard Riemann.. In 1827, Carl Friedrich Gauss discovered that the Gaussian curvature of a surface embedded in 3-dimensional space only depends on local measurements made within the surface (the first fundamental form). [1]
The geometry and topology of three-manifolds is a set of widely circulated notes for a graduate course taught at Princeton University by William Thurston from 1978 to 1980 describing his work on 3-manifolds. They were written by Thurston, assisted by students William Floyd and Steven Kerchoff. [1]
Hyperbolic 3-manifolds of finite volume have a particular importance in 3-dimensional topology as follows from Thurston's geometrisation conjecture proved by Perelman. The study of Kleinian groups is also an important topic in geometric group theory .
The manifolds / with Γ cyclic are precisely the 3-dimensional lens spaces.A lens space is not determined by its fundamental group (there are non-homeomorphic lens spaces with isomorphic fundamental groups); but any other spherical manifold is.
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