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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. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an affine connection, the most elementary type of connection, gives a means for transporting tangent vectors to a manifold from one point to another along a curve. An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields: the infinitesimal transport of a vector field in a given direction. Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point and 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 by specifying the allowed directions of motion of the field. A Koszul connection is a connection generalizing the derivative in a vector bundle. Connections also lead to convenient formulations of ''geometric invariants'', such as the curvature (see also curvature tensor and curvature form), and torsion tensor. ==Motivation: the unsuitability of coordinates== 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. A simple mechanical device that can be used to visualize parallel transport is the south-pointing chariot. 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 : cover a neighborhood ''U''0 of the north pole and ''U''1 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: : Let us now represent a vector field in terms of its components relative to the coordinate derivatives. If ''P'' is a point of ''U''0 ⊂ ''S'', then a vector field may be represented by the pushforward : 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 ''U''0 ∩ ''U''1, it is possible to represent the same vector field with respect to the φ1 coordinates: : To relate the components v0 and v1, apply the chain rule to the identity φ1 = φ0 o φ01: : Applying both sides of this matrix equation to the component vector v1(φ1−1(''P'')) and invoking (1) and (2) yields : 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 ''U''1 coordinate system. That is, the functions v1(''φ''1−1(''P''(''t''))) are constant. However, applying the product rule to (3) and using the fact that ''d''v1/''dt'' = 0 gives : 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Connection (mathematics)」の詳細全文を読む スポンサード リンク
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