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The Geometric Topology
In mathematics, the geometric topology is a topological space, topology one can put on the set ''H'' of hyperbolic 3-manifolds of finite volume. Use Convergence in this topology is a crucial ingredient of hyperbolic Dehn surgery, a fundamental tool in the theory of hyperbolic 3-manifolds. Definition The following is a definition due to Troels Jorgensen: :A sequence \ in ''H'' converges to ''M'' in ''H'' if there are :* a sequence of positive real numbers \epsilon_i converging to 0, and :* a sequence of (1+\epsilon_i)-bi-Lipschitz diffeomorphisms \phi_i: M_ \rightarrow M_{[\epsilon_i, \infty)}, :where the domains and ranges of the maps are the \epsilon_i-thick parts of either the M_i's or ''M''. Alternate definition There is an alternate definition due to Mikhail Gromov (mathematician), Mikhail Gromov. Gromov's topology utilizes the Gromov-Hausdorff metric and is defined on ''pointed'' hyperbolic 3-manifolds. One essentially considers better and better bi-Lipschitz h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
