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Clifford Torus
In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles ''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if ''S'' and ''S'' each exists in its own independent embedding space R and R, the resulting product space will be R4 rather than R3. The historically popular view that the cartesian product of two circles is an R3 torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis ''z'' available to it after the first circle consumes ''x'' and ''y''. Stated another way, a torus embedded in R3 is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R4. The relationship is similar to that of projecting the edges of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors often have different definitions. Formal definition In the following we assume all manifolds are differentiable manifolds of class ''C''''r'' for a fixed , and all morphisms are differentiable of class ''C''''r''. Immersed submanifolds An immersed submanifold of a manifold ''M'' is the image ''S'' of an immersion map ; in general this image will not be a submanifold as a subset, and an immersion map need not even be injective (one-to-one) – it can have self-intersections. More narrowly, one can require that the map be an injection (one-to-one), in which we call it an injective immersion, and define an immersed submanifold to be the image subset ''S'' together with a topology and differe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simon Brendle
Simon Brendle (born June 1981) is a German mathematician working in differential geometry and nonlinear partial differential equations. He received his Dr. rer. nat. from Tübingen University under the supervision of Gerhard Huisken (2001). He was a professor at Stanford University (2005–2016), and is currently a professor at Columbia University. He has held visiting positions at MIT, ETH Zürich, Princeton University, and Cambridge University. Contributions to mathematics Simon Brendle has solved major open problems regarding the Yamabe equation in conformal geometry. This includes his counterexamples to the compactness conjecture for the Yamabe problem, and the proof of the convergence of the Yamabe flow in all dimensions (conjectured by Richard Hamilton). In 2007, he proved the differentiable sphere theorem (in collaboration with Richard Schoen), a fundamental problem in global differential geometry. In 2012, he proved the Hsiang–Lawson's conjecture, a longstand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on consists of a metric tensor at each point of that varies smoothly with . A metric tensor is ''positive-definite'' if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying ''infinitesimal'' distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the distance between and can be defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see minimal surface of revolution): the standard definitions only relate to a local optimum, not a global optimum. Definitions Minimal surfaces can be defined in several equivalent ways in R3. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hsiang–Lawson's Conjecture
In mathematics, Lawson's conjecture states that the Clifford torus is the only minimally embedded torus in the 3-sphere ''S''3. The conjecture was featured by the Australian Mathematical Society Gazette as part of the ''Millennium Problems'' series. In March 2012, Simon Brendle Simon Brendle (born June 1981) is a German mathematician working in differential geometry and nonlinear partial differential equations. He received his Dr. rer. nat. from Tübingen University under the supervision of Gerhard Huisken (2001). He ... gave a proof of this conjecture, based on maximum principle techniques. References Geometric topology Theorems in differential geometry Conjectures that have been proved Theorems in topology {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian Submanifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Motivation Symplectic manifolds arise from classical mechanics; in particular, they are a generalization of the phase space of a closed system. In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold. The term "symplectic", introduced by Weyl, is a calque of "complex"; previously, the "symplectic group" had been called the "line complex group". "Complex" comes from the Latin ''com-plexus'', meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek ''sym-plektikos'' (συμπλεκτικός); in both cases the stem comes from the Indo-European root *pleḱ- The name reflects the deep connections between complex and symplectic structures. By Darboux's Theorem, symplectic manifolds are isomorphic to the standard symplectic vector space locally, hence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stereographic Projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter through the point. It is a smooth, bijective function from the entire sphere except the center of projection to the entire plane. It maps circles on the sphere to circles or lines on the plane, and is conformal, meaning that it preserves angles at which curves meet and thus locally approximately preserves shapes. It is neither isometric (distance preserving) nor equiareal (area preserving). The stereographic projection gives a way to represent a sphere by a plane. The metric induced by the inverse stereographic projection from the plane to the sphere defines a geodesic distance between points in the plane equal to the spherical distance between the spherical points they represent. A two-dimensional coordinate system on the stere ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action (mathematics)
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Transformation
In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we have : \langle u,v \rangle = \langle Tu,Tv \rangle \, . Since the lengths of vectors and the angles between them are defined through the inner product, orthogonal transformations preserve lengths of vectors and angles between them. In particular, orthogonal transformations map orthonormal bases to orthonormal bases. Orthogonal transformations are injective: if Tv = 0 then 0 = \langle Tv,Tv \rangle = \langle v,v \rangle, hence v = 0, so the kernel of T is trivial. Orthogonal transformations in two- or three- dimensional Euclidean space are stiff rotations, reflections, or combinations of a rotation and a reflection (also known as improper rotations). Reflections are transformations that reverse the direction front to back, orthogonal to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Group
In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted . It consists of all orthogonal matrices of determinant . This group is also called the rotation group, generalizing the fact that in dimensions 2 and 3, its elements are the usual r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
