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. Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point to the local geometry at another point. Differential geometry embraces several variations on the connection theme, which fall into two major groups: the infinitesimal and the local theory. The local theory concerns itself primarily with notions of parallel transport and holonomy. The infinitesimal theory concerns itself with the differentiation of geometric data. Thus a covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. A Cartan connection is a way of formulating some aspects of connection theory using differential forms and Lie groups. An Ehresmann connection is a connection in a fibre bundle or a principal bundle using osculating spaces of the derivative of a field. A Koszul connection is a connection generalizing the derivative in a vector bundle. Table of Geometry, from the 1728 Cyclopaedia. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ... In differential geometry, the holonomy group of a connection on a vector bundle over a smooth manifold M is the group of linear transformations induced by parallel transport around closed loops in M. There is an analogous notion for connections on principal bundles over M. The holonomy group of a... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In mathematics, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ... 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. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In differential geometry, an Ehresmann connection is a version of the connection concept which applies to arbitrary fibre bundles. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ... 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). ...

Connections also lead to convenient formulations of geometric invariants, such as the curvature (see also curvature tensor and curvature form), and torsion tensor. Curvature refers to a number of loosely related concepts in different areas of geometry. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... In differential geometry, the curvature form describes curvature of principal bundle with connection. ... In differential geometry, the torsion tensor is one of the tensorial invariants of a connection on the tangent bundle. ...

General concept: the unsuitability of coordinates

Parallel transport on a sphere.

Consider the following problem. Suppose that a tangent vector to the sphere S is given at the north pole, and we are to define a manner of consistently moving this vector to other points of the sphere: a means for parallel transport. Naïvely, this could be done using a particular coordinate system. However, unless proper care is applied, the parallel transport defined in one system of coordinates will not agree with that of another coordinate system. A more appropriate parallel transportation system exploits the symmetry of the sphere under rotation. Given a vector at the north pole, one can transport this vector along a curve by rotating the sphere in such a way that the north pole moves along the curve without axial rolling. This latter means of parallel transport is the Levi-Civita connection on the sphere. If two different curves are given with the same initial and terminal point, and a vector v is rigidly moved along the first curve by a rotation, the resulting vector at the terminal point will be different from the vector resulting from rigidly moving v along the second curve. This phenomenon reflects the curvature of the sphere. Image File history File links Connection-on-sphere. ... 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. ... 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). ... Curvature refers to a number of loosely related concepts in different areas of geometry. ...

For instance, suppose that S is given coordinates by the stereographic projection. Regard S as consisting of unit vectors in R³. Then S carries a pair of coordinate patches: one covering a neighborhood of the north pole, and the other of the south pole. The mappings Stereographic projection of a circle of radius R onto the x axis. ...

cover a neighborhood U₀ of the north pole and U₁ of the south pole, respectively. Let X, Y, Z be the ambient coordinates in R³. Then φ₀ and φ₁ have inverses

,

so that the coordinate transition function is inversion in the circle: In geometry, an inversion is a transformation that maps all circles into circles, where by a circle one may also mean a line (a circle with infinite radius). ...

Let us now represent a vector field in terms of its components relative to the coordinate derivatives. If P is a point of U₀ ⊂ S, then a vector field may be represented by

(1)

where denotes the Jacobian matrix of φ₀, and v₀=v₀(x,y) is a vector field on R² uniquely determined by v. Furthermore, on the overlap between the coordinate charts U₀ ∩ U₁, it is possible to represent the same vector field with respect to the φ₁ coordinates: In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

(2).

To relate the components v₀ and v₁, apply the chain rule to the identity φ₁ = φ₀ o φ₀₁: In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

.

Applying both sides of this matrix equation to the component vector v₁(φ₁^-1(P)) and invoking (1) and (2) yields

. (3)

We come now to the main question of defining how to transport a vector field parallelly along a curve. Suppose that P(t) is a curve in S. Naïvely, one may consider a vector field parallel if the coordinate components of the vector field are constant along the curve. However, an immediate ambiguity arises: in which coordinate system should these components be constant?

For instance, suppose that v(P(t)) has constant components in the U₁ coordinate system. That is, the functions v₁(φ₁^-1(P(t))) are constant. However, applying the product rule to (3) and using the fact that dv₁/dt = 0 gives In mathematics, the product rule of calculus, also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ...

.

But is always a non-singular matrix (provided that the curve P(t) is not stationary), so v₁ and v₀ cannot ever be simultaneously constant along the curve.

Resolution

The problem observed above is that the usual directional derivative of vector calculus does not behave well under changes in the coordinate system when applied to the components of vector fields. This makes it quite difficult to describe how to parallelly translate vector fields, if indeed such a notion makes any sense at all. There are two fundamentally different ways of resolving this problem. In mathematics, the directional derivative of a multivariate differentiable function along a given vector intuitively represents the rate of change of the function in the direction of that vector. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ...

The first approach is to examine what is required for a generalization of the directional derivative to "behave well" under coordinate transitions. This is the tactic taken by the covariant derivative approach to connections: good behavior is equated with covariance. Here one considers a modification of the directional derivative by a certain linear operator, whose components are called the Christoffel symbols, which involves no derivatives on the vector field itself. The directional derivative D_uv of the components of a vector v in a coordinate system φ in the direction u are replaced by a covariant derivative: In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In category theory, see covariant functor. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...

where Γ depends on the coordinate system φ and is bilinear in u and v. In particular, Γ does not involve any derivatives on u or v. In this approach, Γ must transform in a prescribed manner when the coordinate system φ is changed to a different coordinate system. This transformation is not tensorial, since it involves not only the first derivative of the coordinate transition, but also its second derivative. Specifying the transformation law of Γ is not sufficient to determine Γ uniquely. Some other normalization conditions must be imposed, usually depending on the type of geometry under consideration. In Riemannian geometry, the Levi-Civita connection requires compatibility of the Christoffel symbols with the metric (as well as a certain symmetry condition). With these normalizations, the connection is uniquely defined. In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... 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 differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... 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 and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ... 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. ...

The second approach is to use Lie groups to attempt to capture some vestige of symmetry on the space. This is the approach of Cartan connections. The example above using rotations to specify the parallel transport of vectors on the sphere is very much in this vein. In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan. ...

Historical survey of connections

Historically, connections were studied from an infinitesimal perspective in Riemannian geometry. The infinitesimal study of connections began to some extent with Christoffel. This was later taken up more thoroughly by Gregorio Ricci-Curbastro and Tullio Levi-Civita^[1] who observed in part that a connection in the infinitesimal sense of Christoffel also allowed for a notion of parallel transport. In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... Elwin Bruno Christoffel (November 10, 1829 _ March 15, 1900) was a German and French mathematician and physicist. ... Gregorio Ricci-Curbastro (January 12, 1853 - August 6, 1925) was an Italian mathematician. ... Tullio Levi-Civita. ... In mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay parallel with respect to the given connection. ...

The work of Levi-Civita focused exclusively on regarding connections as a kind of differential operator whose parallel displacements were then the solutions of differential equations. As the twentieth century progressed, Elie Cartan developed a new notion of connection. He sought to apply the techniques of Pfaffian systems to the geometries of Felix Klein's erlangen program. In these investigations, he found that a certain infinitesimal notion of connection (a Cartan connection) could be applied to these geometries and more: his connection concept allowed for the presence of curvature which would otherwise be absent in a classical Klein geometry. (See, for example, ^[2], ^[3]) Furthermore, using the dynamics of Darboux, Cartan was able to generalize notion of parallel transport for his class of infinitesimal connections. This established another major thread in the theory of connections: that a connection is a certain kind of differential form. In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... An illustration of a differential equation. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ... In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ... Felix Christian Klein (April 25, 1849, Düsseldorf, Germany – June 22, 1925, Göttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. ... An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan. ... Curvature refers to a number of loosely related concepts in different areas of geometry. ... Jean Gaston Darboux (August 14, 1842, Nîmes â€“ February 23, 1917, Paris) was a French mathematician. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

The two threads in connection theory have persisted through the present day: a connection as a differential operator, and a connection as a differential form. In 1950, Jean-Louis Koszul gave an algebraic framework for regarding a connection as a differential operator by means of the Koszul connection^[4]. The Koszul connection was both more general than that of Levi-Civita, and was easier to work with because it finally was able to eliminate (or at least to hide) the awkward Christoffel symbols from the connection formalism. The attendant parallel displacement operations also had natural algebraic interpretations in terms of the connection. Koszul's definition was subsequently adopted by most of the differential geometry community, since it effectively converted the analytic correspondence between covariant differentiation and parallel translation to an algebraic one. 1950 (MCML) was a common year starting on Sunday (link will take you to calendar). ... Jean-Louis Koszul (born January 3, 1920 in Strasbourg, France) is a mathematician best known for studying geometry and discovering the Koszul complex. ... In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ... In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...

In that same year, Charles Ehresmann, a student of Cartan's, presented a variation on the connection as a differential form view in the context of principal bundles and, more generally, fibre bundles^[5]. Ehresmann connections were, strictly speaking, not a generalization of Cartan connections. Cartan connections were quite rigidly tied to the underlying differential topology of the manifold because of their relationship with Cartan's equivalence method. Ehresmann connections were rather a solid framework for viewing the foundational work of other geometers of the time, such as Shiing-Shen Chern, who had already begun moving away from Cartan connections to study what might be called gauge connections. In Ehresmann's point of view, a connection in a principal bundle consists of a specification of horizontal and vertical vector fields on the total space of the bundle. A parallel translation is then a lifting of a curve from the base to a curve in the principal bundle which is horizontal. This viewpoint has proven especially valuable in the study of holonomy. Charles Ehresmann (1905-1979) was a French mathematician who worked on differential topology and category theory. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In differential geometry, an Ehresmann connection is a version of the connection concept which applies to arbitrary fibre bundles. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, Cartans equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism. ... Chen Xingshen Shiing-Shen Chern (陳省身; pinyin: Chén Xǐngshēn; October 26, 1911 – December 3, 2004) was a Chinese-American mathematician, one of the leading differential geometers of the twentieth century. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In differential geometry, the holonomy group of a connection on a vector bundle over a smooth manifold M is the group of linear transformations induced by parallel transport around closed loops in M. There is an analogous notion for connections on principal bundles over M. The holonomy group of a...

Possible approaches

• A rather direct module-style approach to covariant differentiation, stating the conditions allowing vector fields to act as differential operators on vector bundle sections. See covariant derivative, and Koszul connection for the most general approach along these lines.
• Traditional index notation specifies the connection by components; see Christoffel symbols. (Note: this has three indices, but is not a tensor).
• In pseudo-Riemannian and Riemannian geometry there is a way of deriving a connection from the metric tensor. See Levi-Civita connection.
• The connections referred to above are linear or affine connections. There is also a concept of projective connection, of which the Schwarzian derivative in complex analysis is an instance.
• Using principal bundles, a connection can be realized as a Lie algebra-valued differential forms. See connection form and Cartan connection.
• The most abstract approach may be that suggested by Alexander Grothendieck, where a Grothendieck connection is seen as descent data from infinitesimal neighbourhoods of the diagonal.^[6]

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... 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, a differential operator is a linear operator defined as a function of the differentiation operator. ... 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, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829-1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ... 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 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 differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ... 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). ... An affine connection is a connection on the tangent bundle of a differentiable manifold. ... An affine connection is a connection on the tangent bundle of a differentiable manifold. ... In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold. ... In mathematics, the Schwarzian derivative is a certain operator that is invariant under all linear fractional transformations. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. ... In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan. ... Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century. ... In algebraic geometry and synthetic differential geometry, a Grothendieck connection is a way of viewing connections in terms of descent data from infinitesimal neighbourhoods of the diagonal. ... In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of gluing in topology. ... In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology. ...

References

1. ^ Levi-Civita, T. and Ricci, G. "Méthodes de calcul différential absolu et leurs applications", Math. Ann. B, 54 (1900) 125-201.
2. ^ Cartan, E. "Sur les varietes a connexion projective", Bulletin de la Société Mathématique, 52 (1924) 205-241.
3. ^ Cartan, E., Geometry of Riemannian spaces, Math Sci Press, 1983.
4. ^ Koszul, J. L. "Homologie et cohomologie des algebres de Lie", Bulletin de la Société Mathématique 78 (1950) 65-127.
5. ^ Ehresmann, C. "Les connexions infinitésimales dans un espace fibré différentiable", Colloque de Toplogie, Bruxelles (1950) 29-55.
6. ^ Osserman, B., "Connections, curvature, and p-curvature", preprint.