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Essential Manifold
In geometry, an essential manifold is a special type of closed manifold. The notion was first introduced explicitly by Mikhail Gromov. Definition A closed manifold ''M'' is called essential if its fundamental class 'M''defines a nonzero element in the homology of its fundamental group , or more precisely in the homology of the corresponding Eilenberg–MacLane space ''K''(, 1), via the natural homomorphism :H_n(M)\to H_n(K(\pi,1)), where ''n'' is the dimension of ''M''. Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise. Examples *All closed surfaces (i.e. 2-dimensional manifolds) are essential with the exception of the 2-sphere ''S2''. *Real projective space ''RPn'' is essential since the inclusion *:\mathbb^n \to \mathbb^\infty :is injective in homology, where ::\mathbb^\infty = K(\mathbb_2, 1) :is the Eilenberg–MacLane space of the finite cyclic group of order 2. *All comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lens Space
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions. In the 3-manifold case, a lens space can be visualized as the result of gluing two solid tori together by a homeomorphism of their boundaries. Often the 3-sphere and S^2 \times S^1, both of which can be obtained as above, are not counted as they are considered trivial special cases. The three-dimensional lens spaces L(p;q) were introduced by Heinrich Tietze in 1908. They were the first known examples of 3-manifolds which were not determined by their homology and fundamental group alone, and the simplest examples of closed manifolds whose homeomorphism type is not determined by their homotopy type. J. W. Alexander in 1919 showed that the lens spaces L(5;1) and L(5;2) were not homeomorphic even though they have isomorphic fundamental groups and the same homology, though they do not have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not self-crossing curves such as a figure 8. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Systolic Geometry
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations. See also Introduction to systolic geometry. The notion of systole The ''systole'' of a compact metric space ''X'' is a metric invariant of ''X'', defined to be the least length of a noncontractible loop in ''X'' (i.e. a loop that cannot be contracted to a point in the ambient space ''X''). In more technical language, we minimize length over free loops representing nontrivial conjugacy classes in the fundamental group of ''X''. When ''X'' is a graph, the invariant is usually referred to as the girth, ever since the 1947 article on girth by W. T. Tutte. Possibly inspired by Tutte's article, Loewner started thinking about systolic questions on surfaces in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov's Systolic Inequality For Essential Manifolds
In the mathematical field of Riemannian geometry, M. Gromov's systolic inequality bounds the length of the shortest non-contractible loop on a Riemannian manifold in terms of the volume of the manifold. Gromov's systolic inequality was proved in 1983;see it can be viewed as a generalisation, albeit non-optimal, of Loewner's torus inequality and Pu's inequality for the real projective plane. Technically, let ''M'' be an essential Riemannian manifold of dimension ''n''; denote by sys''π''1(''M'') the homotopy 1-systole of ''M'', that is, the least length of a non-contractible loop on ''M''. Then Gromov's inequality takes the form : \left(\operatorname_1(M)\right)^n \leq C_n \operatorname(M), where ''C''''n'' is a universal constant only depending on the dimension of ''M''. Essential manifolds A closed manifold is called ''essential'' if its fundamental class defines a nonzero element in the homology of its fundamental group, or more precisely in the homology of the cor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces. More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of a connected sum on knots, called the knot sum or composition of knots. Connected sum at a point A connected sum of two ''m''-dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres. If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation. Although the construction uses the choice of the balls, the result is unique up to homeomorphism. One can also make this operation work in the smooth categor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Manifold
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, respectively. In these dimensions, they are important because most manifolds can be made into a hyperbolic manifold by a homeomorphism. This is a consequence of the uniformization theorem for surfaces and the geometrization theorem for 3-manifolds proved by Perelman. Rigorous definition A hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1. Every complete, connected, simply-connected manifold of constant negative curvature -1 is isometric to the real hyperbolic space \mathbb^n. As a result, the universal cover of any closed manifold M of constant negative curvature -1 is \mathbb^n. Thus, every such M can be written as \mathbb^n/\Gamma where \Gamma is a torsion-free discrete group of isometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Gromov (mathematician)
Mikhael Leonidovich Gromov (also Mikhail Gromov, Michael Gromov or Misha Gromov; ; born 23 December 1943) is a Russian-French mathematician known for his work in geometry, analysis and group theory. He is a permanent member of Institut des Hautes Études Scientifiques in France and a professor of mathematics at New York University. Gromov has won several prizes, including the Abel Prize in 2009 "for his revolutionary contributions to geometry". Early years, education and career Mikhail Gromov was born on 23 December 1943 in Boksitogorsk, Soviet Union. His father Leonid Gromov was Russian-Slavic and his mother Lea was of Jewish heritage. Both were pathologists. His mother was the cousin of World Chess Champion Mikhail Botvinnik, as well as of the mathematician Isaak Moiseevich Rabinovich. Gromov was born during World War II, and his mother, who worked as a medical doctor in the Soviet Army, had to leave the front line in order to give birth to him. When Gromov was nine years ol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aspherical Manifold , a type of lens assembly used in photography which contains an aspheric lens, sometimes referred to as ASPH
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Aspherical may refer to: * Aspherical space, a concept in topology * Aspherical lens An aspheric lens or asphere (often labeled ''ASPH'' on eye pieces) is a lens (optics), lens whose surface profiles are not portions of a sphere or Cylinder (geometry), cylinder. In photography, a camera lens, lens assembly that includes an aspheri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–MacLane Space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. (See e.g. ) In this context it is therefore conventional to write the name without a space. is a topological space with a single nontrivial homotopy group. Let ''G'' be a group and ''n'' a positive integer. A connected topological space ''X'' is called an Eilenberg–MacLane space of type K(G,n), if it has ''n''-th homotopy group \pi_n(X) isomorphic to ''G'' and all other homotopy groups trivial. Assuming that ''G'' is abelian in the case that n > 1, Eilenberg–MacLane spaces of type K(G,n) always exist, and are all weak homotopy equivalent. Thus, one may consider K(G,n) as referring to a weak homotopy equivalence class of spaces. It is common to refer to any representative as "a K(G,n)" or as "a model of K(G,n)". Moreover, it is comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
