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In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay ''parallel'' with respect to the connection. The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of ''connecting'' the geometries of nearby points. There may be many notions of parallel transport available, but a specification of one — one way of connecting up the geometries of points on a curve — is tantamount to providing a ''connection''. In fact, the usual notion of connection is the infinitesimal analog of parallel transport. Or, ''vice versa'', parallel transport is the local realization of a connection. As parallel transport supplies a local realization of the connection, it also supplies a local realization of the curvature known as holonomy. The Ambrose-Singer theorem makes explicit this relationship between curvature and holonomy. Other notions of connection come equipped with their own parallel transportation systems as well. For instance, a Koszul connection in a vector bundle also allows for the parallel transport of vectors in much the same way as with a covariant derivative. An Ehresmann or Cartan connection supplies a ''lifting of curves'' from the manifold to the total space of a principal bundle. Such curve lifting may sometimes be thought of as the parallel transport of reference frames. ==Parallel transport on a vector bundle== Let ''M'' be a smooth manifold. Let ''E''→''M'' be a vector bundle with covariant derivative ∇ and ''γ'': ''I''→''M'' a smooth curve parameterized by an open interval ''I''. A section of along ''γ'' is called parallel if : Suppose we are given an element ''e''0 ∈ ''E''''P'' at ''P'' = ''γ''(0) ∈ ''M'', rather than a section. The parallel transport of ''e''0 along ''γ'' is the extension of ''e''0 to a parallel ''section'' ''X'' on ''γ''. More precisely, ''X'' is the unique section of ''E'' along ''γ'' such that # # Note that in any given coordinate patch, (1) defines an ordinary differential equation, with the initial condition given by (2). Thus the Picard–Lindelöf theorem guarantees the existence and uniqueness of the solution. Thus the connection ∇ defines a way of moving elements of the fibers along a curve, and this provides linear isomorphisms between the fibers at points along the curve: : from the vector space lying over γ(''s'') to that over γ(''t''). This isomorphism is known as the parallel transport map associated to the curve. The isomorphisms between fibers obtained in this way will in general depend on the choice of the curve: if they do not, then parallel transport along every curve can be used to define parallel sections of ''E'' over all of ''M''. This is only possible if the curvature of ∇ is zero. In particular, parallel transport around a closed curve starting at a point ''x'' defines an automorphism of the tangent space at ''x'' which is not necessarily trivial. The parallel transport automorphisms defined by all closed curves based at ''x'' form a transformation group called the holonomy group of ∇ at ''x''. There is a close relation between this group and the value of the curvature of ∇ at ''x''; this is the content of the Ambrose-Singer holonomy theorem. 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「Parallel transport」の詳細全文を読む スポンサード リンク
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