 # Differential Geometry

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.

Differentiable manifold. 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 formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a linear space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, their composition on chart intersections in the atlas must be differentiable functions on the corresponding linear space.

History Definition Atlases Charts on a manifold Compatible atlases 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. 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.

It was not until much later that holonomy groups would be used to study Riemannian geometry in a more general setting. 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 . In particular, Supersingular K3 surface. 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 Ha(X,W(k)) are equal to a/2, for all a.

In particular, this gives the standard notion of a supersingular abelian variety. For a variety X over a finite field Fq, it is equivalent to say that the eigenvalues of Frobenius on the l-adic cohomology Ha(X,Ql) are equal to qa/2 times roots of unity. It follows that any variety in positive characteristic whose l-adic cohomology is generated by algebraic cycles is supersingular. A K3 surface whose l-adic cohomology is generated by algebraic cycles is sometimes called a Shioda supersingular K3 surface.

Since the second Betti number of a K3 surface is always 22, this property means that the surface has 22 independent elements in its Picard group (ρ = 22). Another phenomenon which can only occur in positive characteristic is that a K3 surface may be unirational. p2e. Monodromy. Definition Let X be a connected and locally connected based topological space with base point x, and let . For a loop γ: [0, 1] → X based at x, denote a lift under the covering map (starting at a point ) by . Finally, we denote by the endpoint , which is generally different from is exactly , that is, an element [γ] fixes a point in F if and only if it is represented by the image of a loop in based at .

Example These ideas were first made explicit in complex analysis. F(z) = log z then analytic continuation anti-clockwise round the circle |z| = 0.5 will result in the return, not to F(z) but F(z)+2πi. In this case the monodromy group is infinite cyclic and the covering space is the universal cover of the punctured complex plane. Differential equations in the complex domain One important application is to differential equations, where a single solution may give further linearly independent solutions by analytic continuation. . Topological and geometric aspects .

See also Morse theory. "Morse function" redirects here. In another context, a "Morse function" can also mean an anharmonic oscillator: see Morse potential The analogue of Morse theory for complex manifolds is Picard–Lefschetz theory. Basic concepts Contour lines around a saddle point To each of these three types of critical points – basins, passes, and peaks (also called minima, saddles, and maxima) – one associates a number called the index. Intuitively speaking, the index of a critical point b is the number of independent directions around b in which f decreases.

Therefore, the indices of basins, passes, and peaks are 0, 1, and 2, respectively. Define Ma as f−1(−∞, a]. These figures are homotopy equivalent. One therefore appears to have the following rule: the topology of Mα does not change except when α passes the height of a critical point, and when α passes the height of a critical point of index γ, a γ-cell is attached to Mα. This rule, however, is false as stated. Formal development Theorem. . K3 surface. 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 4 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. This means that they cannot be embedded in any projective space as a surface defined by polynomial equations. André Weil (1958) named them in honor of three algebraic geometers, Kummer, Kähler and Kodaira, and the mountain K2 in Kashmir. Definition There are many equivalent properties that can be used to characterize a K3 surface.

From Serre duality. 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. Vector bundles are almost always required to be locally trivial, however, which means they are examples of fiber bundles.

Definition and first consequences Riemannian geometry. 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 (On the Hypotheses which lie at the Bases of Geometry).

It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Introduction Riemannian geometry was first put forward in generality by Bernhard Riemann in the nineteenth century. The following articles provide some useful introductory material: 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 (articles of 1825 and 1827), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Overview Polyhedra in the Euclidean space, such as the boundary of a cube, are among the first surfaces encountered in geometry. It is also possible to define smooth surfaces, in which each point has a neighborhood diffeomorphic to some open set in E2, the Euclidean plane. History of surfaces and. 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. The family 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 Examples Isometries 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.

Description On the other hand, smooth manifolds are more rigid than the topological manifolds. John Milnor discovered that some spheres have more than one smooth structure—see exotic sphere and Donaldson's theorem. Kervaire exhibited topological manifolds with no smooth structure at all. Some constructions of smooth manifold theory, such as the existence of tangent bundles, can be done in the topological setting with much more work, and others cannot.

For a list of differential topology topics, see the following reference: List of differential geometry topics. Differential topology versus differential geometry Differential topology and differential geometry are first characterized by their similarity. See also Notes Special unitary group. 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. Properties The special unitary group SU(n) is a real Lie group (though not a complex Lie group). Its dimension as a real manifold is n2 − 1. Topologically, it is compact and simply connected. Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below). The center of SU(n) is isomorphic to the cyclic group Zn. Its outer automorphism group, for n ≥ 3, is Z2, while the outer automorphism group of SU(2) is the trivial group.

The Lie algebra of SU(n), denoted by su(n) is generated by n2 - 1 operators, which satisfy the commutator relationship for i, j, k, ℓ = 1, 2, ..., n Additionally, the operator satisfies which implies that the number of independent generators is n2 − 1 . Generators and Fundamental representation We also take as a normalization convention. Adjoint representation n = 2 satisfy. 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. Formal definition A Klein geometry is a pair (G, H) where G is a Lie group and H is a closed Lie subgroup of G such that the (left) coset space G/H is connected. Dim X = dim G − dim H. There is a natural smooth left action of G on X given by Given any connected smooth manifold X and a smooth transitive action by a Lie group G on X, we can construct an associated Klein geometry (G, H) by fixing a basepoint x0 in X and letting H be the stabilizer subgroup of x0 in G.

Two Klein geometries (G1, H1) and (G2, H2) are geometrically isomorphic if there is a Lie group isomorphism φ : G1 → G2 so that φ(H1) = H2. Bundle description Examples