# Differential Geometry

Ricci flow Several stages of Ricci flow on a 2D manifold. In differential geometry , the Ricci flow 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 . The Ricci flow was first introduced by Richard Hamilton in 1981, and is also referred to as the Ricci-Hamilton flow . It is the primary tool used in Grigori Perelman's solution of the Poincaré conjecture , as well as in the proof of the Differentiable sphere theorem by Brendle and Schoen. [ edit ] Mathematical definition
A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the middle chart the Tropic of Cancer is a smooth curve, whereas in the first it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable. In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus . Any manifold can be described by a collection of charts , also known as an atlas. Differentiable manifold
Category:Curves
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. In differential geometry , the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported.
In algebraic geometry , a supersingular K3 surface is a K3 surface over a field k of characteristic p > 0 such that the slopes of Frobenius on the crystalline cohomology H 2 ( X , W ( k )) are all equal to 1. [ 1 ] These have also been called Artin supersingular K3 surfaces. Supersingular K3 surfaces can be considered the most special and interesting of all K3 surfaces. [ edit ] Conjectures More generally, a smooth projective variety X over a field of characteristic p > 0 is called supersingular if all slopes of Frobenius on the crystalline cohomology H a ( X , W ( k )) are equal to a /2, for all a . In particular, this gives the standard notion of a supersingular abelian variety . Supersingular K3 surface
In mathematics , monodromy is the study of how objects from mathematical analysis , algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity . As the name implies, the fundamental meaning of monodromy comes from 'running round singly'. It is closely associated with covering maps and their degeneration into ramification ; the aspect giving rise to monodromy phenomena is that certain functions we may wish to define fail to be single-valued as we 'run round' a path encircling a singularity. The failure of monodromy is best measured by defining a monodromy group : a group of transformations acting on the data that encodes what does happen as we 'run round'. [ edit ] Definition Let X be a connected and locally connected based topological space with base point x , and let Monodromy
Morse theory "Morse function" redirects here. In another context, a "Morse function" can also mean an anharmonic oscillator: see Morse potential In differential topology , Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse , a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology . Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography .
Dans la seconde partie de mon rapport, il s'agit des variétés kählériennes dites K3, ainsi nommées en l'honneur de Kummer , Kähler , Kodaira et de la belle montagne K2 au Cachemire André Weil (1958 , p.546), describing the reason for the name "K3 surface" In mathematics , a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle . In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0. Together with two-dimensional complex tori , they are the Calabi-Yau manifolds of dimension two. Most complex K3 surfaces are not algebraic. K3 surface
Vector bundle In mathematics , a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space , a manifold , or an algebraic variety ): to every point x of the space X we associate (or "attach") a vector space V ( x ) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X . The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V ( x ) = V for all x in X : in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X × V over X . Such vector bundles are said to be trivial .
Elliptic geometry is also sometimes called "Riemannian geometry". Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds , smooth manifolds with a Riemannian metric , i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle , length of curves , surface area , and volume . From those some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugurational lecture Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (English: On the hypotheses on which geometry is based). Riemannian geometry
Differential geometry of surfaces In mathematics , the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric . Surfaces have been extensively studied from various perspectives: extrinsically , relating to their embedding in Euclidean space and intrinsically , reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature , first studied in depth by Carl Friedrich Gauss ( 1825-1827 ), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves .
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
Orbifold This terminology should not be blamed on me. It was obtained by a democratic process in my course of 1976–77. An orbifold is something with many folds; unfortunately, the word “manifold” already has a different definition. I tried “foldamani”, which was quickly displaced by the suggestion of “manifolded”. After two months of patiently saying “no, not a manifold, a manifol dead ,” we held a vote, and “orbifold” won. Thurston (1980 , section 13.2) explaining the origin of the word "orbifold"
Differential topology In mathematics , differential topology is the field dealing with differentiable functions on differentiable manifolds . It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds. [ edit ] Description Differential topology considers the properties and structures that require only a smooth structure on a manifold to be defined. Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology.
The special unitary group of degree n , denoted SU( n ), is the group of n × n unitary matrices with determinant 1. (In general, complex unitary matrices have complex determinants with modulus 1, but arbitrary phase .) The group operation is that of matrix multiplication . The special unitary group is a subgroup of the unitary group U( n ), consisting of all n × n unitary matrices, which is itself a subgroup of the general linear group GL( n , C ). The SU( n ) groups find wide application in the Standard Model of particle physics , especially SU(2) in the electroweak interaction and SU(3) in QCD . [ 1 ] The simplest case, SU(1), is the trivial group , having only a single element. Special unitary group
Klein geometry In mathematics , a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program . More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G , which acts as the symmetry group of the geometry. For background and motivation see the article on the Erlangen program .