NationMaster - Encyclopedia: Differential geometry and topology

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It arises naturally from the study of the theory of differential equations. Differential geometry is the study of geometry using differential calculus (cf. integral geometry). These fields are adjacent, and have many applications in physics, notably in the theory of relativity. Together they make up the geometric theory of differentiable manifolds - which can also be studied directly from the point of view of dynamical systems. Image File history File links Derived from public domain images featured at: http://commons. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Partial plot of a function f. ... This article or section is in need of attention from an expert on the subject. ... A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... Calabi-Yau manifold Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... In mathematics, the term integral geometry in is used in two ways, which, although related, imply different views of the content of the subject. ... Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... Two-dimensional analogy of space-time curvature described in General Relativity. ... The Lorenz attractor is an example of a non-linear dynamical system. ...

Differential geometry versus differential topology

Differential topology and differential geometry are first characterized by their similarity. They both study primarily the properties of differentiable manifolds, sometimes with a variety of structures imposed on them.

One major difference lies in the nature of the problems that each subject tries to address. In one view,^[1] differential topology distinguishes itself from differential geometry by studying primarily those problems which are inherently global. Consider the example of a coffee cup and a donut (see this example.) From the point of view of differential topology, the donut and the coffee cup are the same (in a sense). A differential topologist imagines that the donut is made out of a rubber sheet, and that the rubber sheet can be smoothly reshaped from its original configuration as a donut into a new configuration in the shape of a coffee cup without tearing the sheet or gluing bits of it together. This is an inherently global view, though, because there is no way for the differential topologist to tell whether the two objects are the same (in this sense) by looking at just a tiny (local) piece of either of them. She must have access to each entire (global) object. A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...

From the point of view of differential geometry, the coffee cup and the donut are different because it is impossible to rotate the coffee cup in such a way that its configuration matches that of the donut. This is also a global way of thinking about the problem. But an important distinction is that the geometer doesn't need the entire object to decide this. By looking, for instance, at just a tiny piece of the handle, she can decide that the coffee cup is different from the donut because the handle is thinner (or more curved) than any piece of the donut.

To put it succinctly, differential topology studies structures on manifolds which, in a sense, have no interesting local structure. Differential geometry studies structures on manifolds which do have an interesting local (or sometimes even infinitesimal) structure.

More mathematically, for example, the problem of constructing a diffeomorphism between two manifolds of the same dimension is inherently global since locally two such manifolds are always diffeomorphic. Likewise, the problem of computing a quantity on a manifold which is invariant under differentiable mappings is inherently global, since any local invariant will be trivial in the sense that it is already exhibited in the topology of Rⁿ. Moreover, differential topology does not restrict itself necessarily to the study of diffeomorphism. For example, symplectic topology — a subbranch of differential topology — studies global properties of symplectic manifolds. Differential geometry concerns itself with problems — which may be local or global — that always have some non-trivial local properties. Thus differential geometry may study differentiable manifolds equipped with a connection, a metric (which may be Riemannian, pseudo-Riemannian, or Finsler), a special sort of distribution (such as a CR structure), and so on. In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In mathematics, a Finsler manifold is a differential manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth and satisfies the following property: For each point x of M, and for every vector v in the tangent... In mathematics, Frobenius theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations. ... In mathematics, a CR manifold is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge. ...

This distinction between differential geometry and differential topology is blurred, however, in questions specifically pertaining to local diffeomorphism invariants such as the tangent space at a point. Differential topology also deals with questions like this, which specifically pertain to the properties of differentiable mappings on Rⁿ (for example the tangent bundle, jet bundles, the Whitney extension theorem, and so forth). The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x âˆˆ M and v âˆˆ Tx(M), the tangent space... In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain. ... In mathematics, in particular in mathematical analysis, Whitneys extension theorem is a partial converse to Taylors theorem. ...

Nevertheless, the distinction becomes clearer in abstract terms. Differential topology is the study of the (infinitesimal, local, and global) properties of structures on manifolds having no non-trivial local moduli, whereas differential geometry is the study of the (infinitesimal, local, and global) properties of structures on manifolds having non-trivial local moduli.

Intrinsic versus extrinsic

Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves. Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving 'outside' the geometric object because it is considered as given in a free-standing way. Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... An open surface with X-, Y-, and Z-contours shown. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... The ambient space, in mathematics, is the space surrounding a mathematical object. ... In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus. ... Bernhard Riemann. ...

The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic (what would be 'outside' it?). With the intrinsic point of view it is harder to define the central concept of curvature and other structures such as connections, so there is a price to pay. Curvature refers to a number of loosely related concepts in different areas of geometry. ... In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. ...

These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one. (See the Nash embedding theorem.) The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded in a Euclidean space Rn. ...

Technical requirements

The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles, cotangent bundles, differential forms, exterior derivatives, integrals of p-forms over p-dimensional submanifolds and Stokes' theorem, wedge products, and Lie derivatives. These all relate to multivariable calculus; but for geometric applications, differential geometry must be developed in a way that makes good sense without a preferred coordinate system. The distinctive concepts of differential geometry can be said to be those that embody the geometric nature of the second derivative: the many aspects of curvature. On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x âˆˆ M and v âˆˆ Tx(M), the tangent space... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... Multivariable calculus is the extension of calculus in one variable to calculus in several variables: the functions which are differentiated and integrated involve several variables rather than one variable. ... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... Curvature refers to a number of loosely related concepts in different areas of geometry. ...

A real differentiable manifold is a topological space with a collection of diffeomorphisms from open sets of the space to open subsets in Rⁿ such that the open sets cover the space, and if f, g are diffeomorphisms then the composite mapping f o g ^-1 from an open subset of the open unit ball to the open unit ball is infinitely differentiable. We say a function from the manifold to R is infinitely differentiable if its composition with every diffeomorphism results in an infinitely differentiable function from the open unit ball to R. Of course manifolds need not be real, for example we can have complex manifolds. This article or section is in need of attention from an expert on the subject. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... some unit spheres In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used. ...

At every point of the manifold, there is the tangent space at that point, which consists of every possible velocity (direction and magnitude) with which it is possible to travel away from this point. For an n-dimensional manifold, the tangent space at any point is an n-dimensional vector space, or in other words a copy of Rⁿ. The tangent space has many definitions. One definition of the tangent space is as the dual space to the linear space of all functions which are zero at that point, divided by the space of functions which are zero and have a first derivative of zero at that point. Having a zero derivative can be defined by "composition by every differentiable function to the reals has a zero derivative", so it is defined just by differentiability. The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...

A vector field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the tangent bundle), such that at each point, the value is an element of the tangent space at that point. Such a mapping is called a section of a bundle. A vector field is differentiable if for every differentiable function, applying the vector field to the function at each point yields a differentiable function. Vector fields can be thought of as time-independent differential equations. A differentiable function from the reals to the manifold is a curve on the manifold. This defines a function from the reals to the tangent spaces: the velocity of the curve at each point it passes through. A curve will be said to be a solution of the vector field if, at every point, the velocity of the curve is equal to the vector field at that point. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x âˆˆ M and v âˆˆ Tx(M), the tangent space... Look up section in Wiktionary, the free dictionary. ... Bundle may be: a mathematical term referring to generalization of a fiber bundle dropping the condition of a local product structure. ...

An alternating k-dimensional linear form is an element of the antisymmetric k'th tensor power of the dual V^* of some vector space V. A differential k-form on a manifold is a choice, at each point of the manifold, of such an alternating k-form -- where V is the tangent space at that point. This will be called differentiable if whenever it operates on k differentiable vector fields, the result is a differentiable function from the manifold to the reals. A space form is a linear form with the dimensionality of the manifold.

Differential topology

Differential topology per se considers the properties and structures that require only a smooth structure on a manifold to define (such as those in the previous section). 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. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold—that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume.

Conversely, smooth manifolds are more rigid than the topological manifolds. Certain topological manifolds have no smooth structures at all (see Donaldson's theorem) and others have more than one inequivalent smooth structure (such as exotic spheres). 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. In mathematics, Donaldsons theorem states that a positive definite intersection form of a simply connected smooth manifold of dimension 4 is diagonalisable to the unit matrix. ... In mathematics, an exotic sphere is a differential manifold M, such that from a topological point of view M is a sphere, but not from the point of view of its differential structure. ...

See also transversality Transversality in mathematics is a notion that describes how spaces can intersect; transversality can be seen as the opposite of tangency, and plays a role in general position. ...

Branches of differential geometry

Riemannian geometry

Riemannian geometry has Riemannian manifolds as the main object of study — smooth manifolds with additional structure which makes them look infinitesimally like Euclidean space. These allow one to generalise the notion from Euclidean geometry and analysis such as gradient of a function, divergence, length of curves and so on; without assumptions that the space is globally so symmetric. The Riemannian curvature tensor is an important pointwise invariant associated to a Riemannian manifold that measures how close it is to being flat. In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... For other uses, see Gradient (disambiguation). ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... For other uses of this word, see Length (disambiguation). ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... Curvature refers to a number of loosely related concepts in different areas of geometry. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

Pseudo-Riemannian geometry

Pseudo-Riemannian geometry generalizes the flat Minkowski space of Einstein's special relativity to curved space. Pseudo-Riemannian geometry is the mathematical basis of Einstein's general relativity theory of gravity. In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... General relativity (GR) or general relativity theory (GRT) is the theory of gravitation published by Albert Einstein in 1915. ...

Finsler geometry

Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e. a Banach norm defined on each tangent space. A Finsler metric is much more general structure than a Riemannian metric. Then a Finsler structure on a manifold M is a function F : TM → [0,n) such that: In mathematics, a Finsler manifold is a differential manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth and satisfies the following property: For each point x of M, and for every vector v in the tangent... In mathematics, a Finsler manifold is a differential manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth and satisfies the following property: For each point x of M, and for every vector v in the tangent... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

Symplectic geometry

Symplectic geometry is the study of symplectic manifolds. A symplectic manifold is a differentiable manifold equipped with a symplectic form

ω

(that is, a non-degenerate, bilinear, skew-symmetric and closed 2-form). Since the symplectic form must be skew-symmetric, its matrix representation must be skew-symmetric, i.e. In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

It follows that

det(M) = ( - 1) n det(M)

M

is an $n times n$ matrix. Thus, for odd

n

we see that

det(M) = 0,

and so non-degenerate skew-symmetric two forms can only exist on even dimensional spaces. Unlike in Riemannian geometry, all symplectic manifolds are locally isomorphic: this is called Darboux's theorem and follows from the assumption that

ω

is closed, so the only invariants of a symplectic manifold are global in nature. A diffeomorphism between two symplectic spaces which preserves the symplectic structure (i.e. the symplectic form) is called a symplectomorphism. Darbouxs theorem is a theorem in symplectic topology which states that every symplectic manifold (of fixed dimension) is locally symplectomorphic. ... In mathematics, a symplectomorphism (or Hamiltonian flow) is an isomorphism in the category of symplectic manifolds. ...

In dimension 2, a symplectic manifold is just a manifold endowed with an area form. The first result in symplectic topology is probably the Poincaré-Birkhoff theorem, conjectured by Henri Poincaré and proved by George David Birkhoff in 1912. This claims that if an area preserving map of a ring twists each boundary component in opposite directions, then the map has at least two fixed points. Jules TuPac Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... George David Birkhoff George David Birkhoff (21 March 1884, Overisel, Michigan - 12 November 1944, Cambridge, Massachusetts) was an American mathematician, best known for what is now called the ergodic theorem. ...

It is easy to show that the area preserving condition (or the twisting condition) cannot be removed.

Note that if one tries to extend such a theorem to higher dimensions, one would probably guess that a volume preserving map of a certain type must have fixed points. This is false in dimensions greater than 3.

Contact geometry

Contact geometry is an analog of symplectic geometry which works for certain manifolds of odd dimension. Roughly, the contact structure on (2n+1)-dimensional manifold is a choice of a hyperplane field that is nowhere integrable. This is equivalent to the hyperplane field being defined by a 1-form

α

such that $alphawedge (dalpha)^n$ does not vanish anywhere. In mathematics, contact geometry is the study of completely nonintegrable hyperplane fields on manifolds. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

Complex and Kähler geometry

Complex differential geometry is the study of complex manifolds. An almost complex manifold is a real manifold

M

, endowed with a tensor of type (1,1), i.e. a vector bundle endomorphism (called an almost complex structure) In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, an almost complex manifold is a smooth manifold equipped with a structure that, roughly speaking, defines a multiplication by i on each tangent space. ...

It follows from this definition that an almost complex manifold is even dimensional.

An almost complex manifold is called complex if

N J = 0

, where

N J

is a tensor of type (2,1) related to

J

, called the Nijenhuis tensor (or sometimes the torsion). An almost complex manifold is complex if and only if admits an holomorphic coordinate atlas. An almost Hermitian structure is given by an almost complex structure J, along with a riemannian metric g, satisfying the compatibility condition

g (J X, J Y) = g (X, Y)

. An almost hermitian structure defines naturally a differential 2-form

ω J, g (X, Y): = g (J X, Y)

The following two conditions are equivalent: In mathematics, an almost complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... In topology, a branch of mathematics, an atlas describes how a complicated space called a manifold is glued together from simpler pieces. ... In mathematics, a Hermitian manifold is a topological manifold with a metric g which is Hermitian for an almost complex structure J. Definition A metric g is called a Hermitian metric if for all vector fields v, w on M. In index notation, one writes A manifold with such a... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

where $nabla$ is the Levi-Civita connection of

g

. In this case,

(J, g)

is called a Kähler structure, and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold. A large class of Kähler manifolds (the class of Hodge manifolds) is given by all the smooth complex projective varieties. In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ... In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. ... In mathematics, a symplectic manifold is a smooth manifold equipped with a closed, nondegenerate 2-form. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...

CR geometry

CR geometry is the study of the intrinsic geometry of boundaries of domains in complex manifolds. In mathematics, a CR manifold is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ...

Contents