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C and M Connection biography, C and M Connection discography
We are proud of our customers and our work.An affine connection on the sphere rolls the affine tangent plane from one point to another.As it does so, the point of contact traces out a curve in the plane: the development.The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space Rn by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.This also defines a parallel transport on the frame bundle.Affine connections may also be used to define (affine) geodesics on a manifold, generalizing the straight lines of Euclidean space, although the geometry of those straight lines can be very different from usual Euclidean geometry; these differences are essentially encapsulated in the curvature of the connection.Motivation from tensor calculus
1.Parallel transport for affine connections
4 Formal definition on the frame bundle
5 Affine connections as Cartan connections
5.Affine space as the flat model geometry
5.Definition of an affine space
5.Definition as a principal affine connection
5.Smooth functions and vector fields can be defined on manifolds, just as they can on Euclidean space, and scalar functions on manifolds can be differentiated in a natural way.However, differentiation of vector fields is less straightforward: this is a simple matter in Euclidean space, because the tangent space of based vectors at a point p can be identified naturally (by translation) with the tangent space at a nearby point q.The notion of an affine connection was introduced to remedy this problem by connecting nearby tangent spaces.As the tangent plane is rolled on S, the point of contact traces out a curve on S.This provides a way to identify the tangent planes at different points along the curve: in particular, a tangent vector in the tangent space at one point on the curve is identified with a unique tangent vector at any other point on the curve.These identifications are always given by affine transformations from one tangent plane to another.This notion of parallel transport of tangent vectors, by affine transformations, along a curve has a characteristic feature: the point of contact of the tangent plane with the surface always moves with the curve under parallel translation (i.In more modern approaches, the point of contact is viewed as the origin in the tangent plane (which is then a vector space), and the movement of the origin is corrected by a translation, so that parallel transport is linear, rather than affine.Motivation from tensor calculus
See also: covariant derivative
The second motivation for affine connections comes from the notion of a covariant derivative of vector fields.Civita between 1880 and the turn of the 20th century.The tensor calculus really came to life, however, with the advent of Albert Einstein's theory of general relativity in 1915.Approaches
The complex history has led to the development of widely varying approaches to and generalizations of the affine connection concept.The most popular approach is probably the definition motivated by covariant derivatives.On the one hand, the ideas of Weyl were taken up by physicists in the form of gauge theory and gauge covariant derivatives.Louis Koszul, who defined (linear or Koszul) connections on vector bundles.Alternatively, Euclidean space is a principal homogeneous space or torsor under the group of translations, which is a subgroup of the affine group.As mentioned in the introduction, there are several ways to make this precise: one uses the fact that an affine connection defines a notion of parallel transport of vector fields along a curve.This also defines a parallel transport on the frame bundle.M,TM) be the space of vector fields on M, that is, the space of smooth sections of the tangent bundle TM.Parallel transport of a tangent vector along a curve in the sphere.Thus parallel transport provides a way of moving tangent vectors along a curve using the affine connection to keep them "pointing in the same direction" in an intuitive sense, and this provides a linear isomorphism between the tangent spaces at the two ends of the curve.Affine connections as Cartan connections
See also: Cartan connection
Affine connections can be defined within Cartan's general framework.In the modern approach, this is closely related to the definition of affine connections on the frame bundle.Indeed, in one formulation, a Cartan connection is an absolute parallelism of a principal bundle satisfying suitable properties.Cartan connection ultimately identifies with the tangent space.R3, are infinite in extent.However these affine planes all have a marked point, the point of contact with the surface, and they are tangent to the surface at this point.However, the parallel transport defined by rolling does not fix this "origin": it is affine rather than linear; the linear parallel transport can be recovered by applying a translation.To motivate this, it suffices to consider how affine frames of reference transform infinitesimally with respect to parallel transport.This is the origin of Cartan's method of moving frames.The tangential affine space Ax is thus identified intuitively with an infinitesimal affine neighborhood of x.It describes the geometry of points and free vectors in space.However, a vector v may be added to a point p by placing the initial point of the vector at p and then transporting p to the terminal point.By analogy, the affine group Aff(n) is the group of transformations of An preserving the affine structure.T(v)
where T is a general linear transformation.Cartan equations for the Lie group Aff(n) (identified with FA by the choice of a reference frame).General affine geometries: formal definitions
An affine space, as with essentially any smooth Klein geometry, is a manifold equipped with a flat Cartan connection.There are several ways to approach the definition and two will be given.Lie algebra aff(n) of the affine group Aff(n).M, making it into an affine manifold.Aff(n) A, which is a fiber bundle over M whose fiber at x in M is an affine space Ax.The Cartan condition ensures that the distinguished section a always moves under parallel transport.As there are many equivalent ways to define the notion of an affine connection, so there are many different ways to define curvature and torsion.The first expression is called the torsion of the connection, and the second is also called the curvature.However, they are horizontal and equivariant, and hence define tensorial objects.Civita connection, there is a formula for these components in terms of the components of g.R, containing 0, on which the geodesic is defined.Lindelof theorem, and allows for the definition of an exponential map associated to the affine connection.Development
An affine connection defines a notion of development of curves.Intuitively, development captures the notion that if xt is a curve in M, then the affine tangent space at x0 may be rolled along the curve.As it does so, the marked point of contact between the tangent space and the manifold traces out a curve Ct in this affine space: the development of xt.Tx0M be the linear parallel transport map associated to the affine connection.In particular, xt is a geodesic if and only if its development is an affinely parametrized straight line in Tx0M.Surface theory revisited
If M is a surface in R3, it is easy to see that M has a natural affine connection.From the linear connection point of view, the covariant derivative of a vector field is defined by differentiating the vector field, viewed as a map from M to R3, and then projecting the result orthogonally back onto the tangent spaces of M.Civita connection of this metric.Example: the unit sphere in Euclidean space
Let be the usual scalar product on R3, and let S2 be the unit sphere.So all that needs to be proved here is that the map above does indeed define a tangent vector field.The map f is constant, hence its differential vanishes.The equation (1) above follows.As a result, many mathematicians use the term linear connection (instead of affine connection) for a connection on the tangent bundle, on the grounds that parallel transport is linear and not affine.It is difficult to make Cartan's intuition precise without invoking smooth infinitesimal analysis, but one way is to regard his points being variable, that is maps from some unseen parameter space into the manifold, which can then be differentiated.Nomizu (1996) Volume 1, sections 1.Lumiste, Connections on a manifold (2001).Cartan implicitly identifies this with x in M.See also Sharpe (1997) for a thorough discussion of development in other geometrical situations.Primary historical references
Cartan, Elie (1923).Includes a detailed discussion of the physics of reference frames, and how the connection reflects the physical notion of transport along a worldline.Geometry of Riemannian Spaces, (Appendices by Robert Hermann.Affine connections from the point of view of Riemannian geometry.Robert Hermann's appendices discuss the motivation from surface theory, as well as the notion of affine connections in the modern sense of Koszul.Ehlers (1980), translated 4th edition Space Time Matter by Henry Brose, 1922 (Methuen, reprinted 1952 by Dover), Springer, Berlin.Kobayashi, Shoshichi and Nomizu, Katsumi (1996 (New edition)).This is the main reference for the technical details of the article.Volume 1, chapter III gives a detailed account of affine connections from the perspective of principal bundles on a manifold, parallel transport, development, geodesics, and associated differential operators.Volume 1 chapter VI gives an account of affine transformations, torsion, and the general theory of affine geodesy.Volume 2 gives a number of applications of affine connections to homogeneous spaces and complex manifolds, as well as to other assorted topics.Two articles by Lumiste, giving precise conditions on parallel transport maps in order that they define affine connections.Differential Geometry: Cartan's Generalization of Klein's Erlangen Program.Appendix A elucidates the relationship between the principal connection and absolute parallelism viewpoints.This page was last modified 00:00, 19 November 2007.All text is available under the terms of the GNU Free Documentation License.
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