Cayley graph The Cayley graph of the free group on two generators a and b Definition Suppose that is a generating set. is a colored directed graph constructed as follows:  Each element of is assigned a vertex: the vertex set of is identified with Each generator of is assigned a color .For any the vertices corresponding to the elements and are joined by a directed edge of colour Thus the edge set consists of pairs of the form with providing the color. In geometric group theory, the set is usually assumed to be finite, symmetric (i.e. Examples Cayley graph of the dihedral group Dih4 on two generators a and b On two generators of Dih4, which are both self-inverse A Cayley graph of the dihedral group D4 on two generators a and b is depicted to the left. A different Cayley graph of Dih4 is shown on the right. b is still the horizontal reflection and represented by blue lines; c is a diagonal reflection and represented by green lines. Part of a Cayley graph of the Heisenberg group. . .
Hopf fibration The Hopf fibration can be visualized using a stereographic projection of S3 to R3 and then compressing R3 to a ball. This image shows points on S2 and their corresponding fibers with the same color. Pairwise linked keyrings mimic part of the Hopf fibration. In the mathematical field of topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle. Technically, Hopf found a many-to-one continuous function (or "map") from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere comes from a distinct circle of the 3-sphere (Hopf 1931). This fiber bundle structure is denoted meaning that the fiber space S1 (a circle) is embedded in the total space S3 (the 3-sphere), and p : S3 → S2 (Hopf's map) projects S3 onto the base space S2 (the ordinary 2-sphere).
Poincaré disk model Metric If u and v are two vectors in real n-dimensional vector space Rn with the usual Euclidean norm, both of which have norm less than 1, then we may define an isometric invariant by where denotes the usual Euclidean norm. Such a distance function is defined for any two vectors of norm less than one, and makes the set of such vectors into a metric space which is a model of hyperbolic space of constant curvature −1. The associated metric tensor of the Poincaré disk model is given by where the xi are the Cartesian coordinates of the ambient Euclidean space. Relation to the hyperboloid model The hyperboloid model can be seen as the equation of t2=x2+y2+1 can can be used to construct a Poincaré disk model as a perspective projection viewed from (t=-1,x=0,y=0), projecting the upper half hyperboloid onto an (x,y) unit disk at t=0. The Poincaré disk model, as well as the Klein model, are related to the hyperboloid model projectively. Angles Artistic realizations M.
Convergence of Fourier series Determination of convergence requires the comprehension of pointwise convergence, uniform convergence, absolute convergence, Lp spaces, summability methods and the Cesàro mean. Preliminaries Consider ƒ an integrable function on the interval [0,2π]. are defined by the formula It is common to describe the connection between ƒ and its Fourier series by The notation ~ here means that the sum represents the function in some sense. The question we will be interested in is: Do the functions (which are functions of the variable t we omitted in the notation) converge to ƒ and in which sense? Before continuing the Dirichlet kernel needs to be introduced. , inserting it into the formula for and doing some algebra will give that where ∗ stands for the periodic convolution and is the Dirichlet kernel which has an explicit formula, The Dirichlet kernel is not a positive kernel, and in fact, its norm diverges, namely a fact that will play a crucial role in the discussion. If for and and therefore, if Suppose
Schwarz triangle In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere, possibly overlapping, through reflections in its edges. They were classified in (Schwarz 1873). These can be defined more generally as tessellations of the sphere, the Euclidean plane, or the hyperbolic plane. Each Schwarz triangle on a sphere defines a finite group, while on the Euclidean or hyperbolic plane they define an infinite group. A Schwarz triangle is represented by three rational numbers (p q r) each representing the angle at a vertex. The value n/d means the vertex angle is d/n of the half-circle. "2" means a right triangle. Solution space A fundamental domain triangle, (p q r), can exist in different space depending on this constraint: Graphical representation A Schwarz triangle is represented graphically by a triangular graph. Order-2 edges represent perpendicular mirrors that can be ignored in this diagram. A list of Schwarz triangles
Point groups in three dimensions In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices. O(3) itself is a subgroup of the Euclidean group E(3) of all isometries. Symmetry groups of objects are isometry groups. The point groups in three dimensions are heavily used in chemistry, especially to describe the symmetries of a molecule and of molecular orbitals forming covalent bonds, and in this context they are also called molecular point groups. Finite Coxeter groups are a special set of point groups generated purely by a set of reflectional mirrors passing through the same point. Group structure O(3) is the direct product of SO(3) and the group generated by inversion (denoted by its matrix −I): where isometry ( A, I ) is identified with A.
Villarceau circles Conceptual animation showing how a slant cut torus reveals a pair of circles, known as Villarceau circles In geometry, Villarceau circles /viːlɑrˈsoʊ/ are a pair of circles produced by cutting a torus diagonally through the center at the correct angle. Given an arbitrary point on a torus, four circles can be drawn through it. Example For example, let the torus be given implicitly as the set of points on circles of radius three around points on a circle of radius five in the xy plane Slicing with the z = 0 plane produces two concentric circles, x2 + y2 = 22 and x2 + y2 = 82. Two example Villarceau circles can be produced by slicing with the plane 3x = 4z. and The slicing plane is chosen to be tangent to the torus while passing through its center. Existence and equations A proof of the circles’ existence can be constructed from the fact that the slicing plane is tangent to the torus at two points. The cross-section of the swept surface in the xz plane now includes a second circle.
Hadamard space In an Hadamard space, a triangle is hyperbolic; that is, the middle one in the picture. In fact, any complete metric space where a triangle is hyperbolic is an Hadamard space. In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. The point m is then the midpoint of x and y: In a Hilbert space, the above inequality is equality (with ), and in general an Hadamard space is said to be flat if the above inequality is equality. fixes the circumcenter of B. The basic result for a non-positively curved manifold is the Cartan–Hadamard theorem. See also References Jump up ^ The assumption on "nonempty" has meaning: a fixed point theorem often states the set of fixed point is an Hadamard space.
Ricci flow Several stages of Ricci flow on a 2D manifold. In differential geometry, the Ricci flow (/ˈriːtʃi/) is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric. Mathematical definition Given a Riemannian manifold with metric tensor , we can compute the Ricci tensor The normalized Ricci flow makes sense for compact manifolds and is given by the equation where is the average (mean) of the scalar curvature (which is obtained from the Ricci tensor by taking the trace) and is the dimension of the manifold. The factor of −2 is of little significance, since it can be changed to any nonzero real number by rescaling t. Informally, the Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions. Examples If the manifold is Euclidean space, or more generally Ricci-flat, then Ricci flow leaves the metric unchanged.
Riemannian manifold In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real smooth manifold M equipped with an inner product on the tangent space at each point that varies smoothly from point to point in the sense that if X and Y are vector fields on M, then is a smooth function. of inner products is called a Riemannian metric (tensor). A Riemannian metric (tensor) makes it possible to define various geometric notions on a Riemannian manifold, such as angles, lengths of curves, areas (or volumes), curvature, gradients of functions and divergence of vector fields. Introduction In 1828, Carl Friedrich Gauss proved his Theorema Egregium (remarkable theorem in Latin), establishing an important property of surfaces. Overview Smoothness of α(t) for t in [0, 1] guarantees that the integral L(α) exists and the length of this curve is defined. Riemannian manifolds as metric spaces Properties Riemannian metrics defines a smooth function M → R.
Holonomy Parallel transport on a sphere depends on the path. Transporting from A → N → B → A yields a vector different from the initial vector. This failure to return to the initial vector is measured by the holonomy of the connection. The study of Riemannian holonomy has led to a number of important developments. The holonomy was introduced by Cartan (1926) in order to study and classify symmetric spaces. Definitions Holonomy of a connection in a vector bundle The restricted holonomy group based at x is the subgroup coming from contractible loops γ. If M is connected then the holonomy group depends on the basepoint x only up to conjugation in GL(k, R). Choosing different identifications of Ex with Rk also gives conjugate subgroups. Some important properties of the holonomy group include: Holonomy of a connection in a principal bundle such that . , will not generally be p but rather some other point p·g in the fiber over x. The holonomy group of ω based at p is then defined as
Poincaré conjecture By contrast, neither of the two colored loops on this torus can be continuously tightened to a point. A torus is not homeomorphic to a sphere. Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it. The Poincaré conjecture, before being proven, was one of the most important open questions in topology. History Poincaré's question At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology—what would later be called combinatorial topology and then algebraic topology. In the same paper, Poincaré wondered whether a 3-manifold with the homology of a 3-sphere and also trivial fundamental group had to be a 3-sphere. The original phrasing was as follows: Attempted solutions Dimensions Solution
Borromean rings Mathematical properties Although the typical picture of the Borromean rings (above right picture) may lead one to think the link can be formed from geometrically ideal circles, they cannot be. Freedman and Skora (1987) prove that a certain class of links, including the Borromean links, cannot be exactly circular. Alternatively, this can be seen from considering the link diagram: if one assumes that circles 1 and 2 touch at their two crossing points, then they either lie in a plane or a sphere. In either case, the third circle must pass through this plane or sphere four times, without lying in it, which is impossible; see (Lindström & Zetterström 1991). A realization of the Borromean rings as ellipses 3D image of Borromean Rings Linking In knot theory, the Borromean rings are a simple example of a Brunnian link: although each pair of rings is unlinked, the whole link cannot be unlinked. Hyperbolic geometry Connection with braids History Partial rings
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