|
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean derivative along a tangent vector onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component. This article presents an introduction to the covariant derivative of a vector field with respect to a vector field, both in a coordinate free language and using a local coordinate system and the traditional index notation. The covariant derivative of a tensor field is presented as an extension of the same concept. The covariant derivative generalizes straightforwardly to a notion of differentiation associated to a connection on a vector bundle, also known as a Koszul connection. ==Introduction and history== Historically, at the turn of the 20th century, the covariant derivative was introduced by Gregorio Ricci-Curbastro and Tullio Levi-Civita in the theory of Riemannian and pseudo-Riemannian geometry. Ricci and Levi-Civita (following ideas of Elwin Bruno Christoffel) observed that the Christoffel symbols used to define the curvature could also provide a notion of differentiation which generalized the classical directional derivative of vector fields on a manifold.〔; reprint, ed. Weber, H. (1953), New York: Dover.〕 This new derivative – the Levi-Civita connection – was ''covariant'' in the sense that it satisfied Riemann's requirement that objects in geometry should be independent of their description in a particular coordinate system. It was soon noted by other mathematicians, prominent among these being Hermann Weyl, Jan Arnoldus Schouten, and Élie Cartan,〔cf. with 〕 that a covariant derivative could be defined abstractly without the presence of a metric. The crucial feature was not a particular dependence on the metric, but that the Christoffel symbols satisfied a certain precise second order transformation law. This transformation law could serve as a starting point for defining the derivative in a covariant manner. Thus the theory of covariant differentiation forked off from the strictly Riemannian context to include a wider range of possible geometries. In the 1940s, practitioners of differential geometry began introducing other notions of covariant differentiation in general vector bundles which were, in contrast to the classical bundles of interest to geometers, not part of the tensor analysis of the manifold. By and large, these generalized covariant derivatives had to be specified ''ad hoc'' by some version of the connection concept. In 1950, Jean-Louis Koszul unified these new ideas of covariant differentiation in a vector bundle by means of what is known today as a Koszul connection or a connection on a vector bundle. Using ideas from Lie algebra cohomology, Koszul successfully converted many of the analytic features of covariant differentiation into algebraic ones. In particular, Koszul connections eliminated the need for awkward manipulations of Christoffel symbols (and other analogous non-tensorial) objects in differential geometry. Thus they quickly supplanted the classical notion of covariant derivative in many post-1950 treatments of the subject. 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「covariant derivative」の詳細全文を読む スポンサード リンク
|