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Parallelizable
In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist Smooth function, smooth vector fields \ on the manifold, such that at every point p of M the tangent vectors \ provide a Basis of a vector space, basis of the tangent space at p. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of frame bundle, linear frames has a global section on M. A particular choice of such a basis of vector fields on M is called a Parallelization (mathematics), parallelization (or an absolute parallelism) of M. Examples *An example with n = 1 is the circle: we can take ''V''1 to be the unit tangent vector field, say pointing in the anti-clockwise direction. The torus of dimension n is also parallelizable, as can be seen by expressing it as a cartesian product of circles. For example, take n = 2, and construct a torus from a square of graph paper with opposite edges glued together, to get an idea of the two tangen ...
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Tangent Bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of manifold the tangent spaces and have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle , see Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle. of the tangent spaces of M . That is, : \begin TM &= \bigsqcup_ T_xM \\ &= \bigcup_ \left\ \times T_xM \\ &= \bigcup_ \left\ \\ &= \left\ \end where T_x M denotes the tangent space to M at the point x . So, an el ...
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Frame Bundle
In mathematics, a frame bundle is a principal fiber bundle F(E) associated with any vector bundle ''E''. The fiber of F(E) over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E_x''. The general linear group acts naturally on F(E) via a change of basis, giving the frame bundle the structure of a principal ''\mathrm(k,\mathbb)''-bundle (where ''k'' is the rank of ''E''). The frame bundle of a smooth manifold is the one associated with its tangent bundle. For this reason it is sometimes called the tangent frame bundle. Definition and construction Let ''E \to X'' be a real vector bundle of rank ''k'' over a topological space ''X''. A frame at a point ''x \in X'' is an ordered basis for the vector space ''E_x''. Equivalently, a frame can be viewed as a linear isomorphism :p : \mathbf^k \to E_x. The set of all frames at ''x'', denoted ''F_x'', has a natural right action by the general linear group ''\mathrm(k,\mathbb)'' of invertible ''k \times k'' matrices: ...
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Raoul Bott
Raoul Bott (September 24, 1923 – December 20, 2005) was a Hungarian-American mathematician known for numerous foundational contributions to geometry in its broad sense. He is best known for his Bott periodicity theorem, the Morse–Bott functions which he used in this context, and the Borel–Bott–Weil theorem. Early life Bott was born in Budapest, Kingdom of Hungary (1920–1946), Hungary, the son of Margit Kovács and Rudolph Bott. His father was of Austrian descent, and his mother was of Hungarian Jewish descent; Bott was raised a Catholic by his mother and stepfather in Bratislava, Czechoslovakia, now the capital of Slovakia. Bott grew up in Czechoslovakia and spent his working life in the United States. His family emigrated to Canada in 1938, and subsequently he served in the Canadian Forces, Canadian Army in Europe during World War II. Career Bott later went to college at McGill University in Montreal, where he studied electrical engineering. He then earned a PhD in math ...
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ...
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Hairy Ball Theorem
The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous function, continuous tangent vector field on even-dimensional n‑sphere, ''n''-spheres. For the ordinary sphere, or 2‑sphere, if ''f'' is a continuous function that assigns a Vector (geometric), vector in to every point ''p'' on a sphere such that ''f''(''p'') is always tangent to the sphere at ''p'', then there is at least one pole, a point where the field vanishes (a ''p'' such that ''f''(''p'') = Null vector, 0). The theorem was first proved by Henri Poincaré for the 2-sphere in 1885, and extended to higher even dimensions in 1912 by Luitzen Egbertus Jan Brouwer. The theorem has been expressed colloquially as "you can't comb a hairy ball flat without creating a cowlick" or "you can't comb the hair on a coconut". Counting zeros Every zero of a vector field has a (non-zero) "Vector field#Index of a vector field, index", and it ca ...
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Closed Manifold
In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only Connected space, connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RP''n'' is a closed ''n''-dimensional manifold. The complex projective space CP''n'' is a closed 2''n''-dimensional manifold. A Real line, line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Properties Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups. If M is a closed connected n-manifold, the n-th homology group H_(M;\mathbb) is \mathbb or 0 depending on whether M is Orientability, orientable or not. Moreover, ...
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Orientability
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds oft ...
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Manifold
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 Neighbourhood (mathematics), neighborhood that is homeomorphic to an open (topology), open subset of n-dimensional Euclidean space. One-dimensional manifolds include Line (geometry), lines and circles, but not Lemniscate, self-crossing curves such as a figure 8. Two-dimensional manifolds are also called Surface (topology), surfaces. Examples include the Plane (geometry), 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 ...
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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 ...
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Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have eight dimension (vector space), dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are commutative property, noncommutative and associative property, nonassociative, but satisfy a weaker form of associativity; namely, they are alternative algebra, alternative. They are also Power associativity, power associative. Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Octonions are related to exceptional structures in mathematics, among them the Simple Lie group#Exceptional cases, exceptional Lie groups. Octonions have applications in fields such as string theory, special relativity and qu ...
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Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by (for ''Hamilton''), or in blackboard bold by \mathbb H. Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form a + b\,\mathbf i + c\,\mathbf j +d\,\mathbf k, where the coefficients , , , are real numbers, and , are the ''basis vectors'' or ''basis elements''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, robotics, magnetic resonance i ...
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Normed Division Algebra
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a nondegenerate positive-definite quadratic form. The theorem states that if the quadratic form defines a homomorphism into the positive real numbers on the non-zero part of the algebra, then the algebra must be isomorphic to the real numbers, the complex numbers, the quaternions, or the octonions, and that there are no other possibilities. Such algebras, sometimes called Hurwitz algebras, are examples of composition algebras. The theory of composition algebras has subsequently been generalized to arbitrary quadratic forms and arbitrary fields. Hurwitz's theorem implies that multiplicative formulas for sums of squares can only occur in 1, 2, 4 and 8 dimensions, a result originally proved by Hurwitz in 1898. It is a special case of the Hurwitz problem, solved also i ...
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