A "naive" attempt to define the derivative of a tensor field with respect to a vector field would be to take the directional derivative of the components of the tensor field with respect to the vector field. However, this definition is undesirable because it is not invariant under coordinate transformations, and is thus meaningless when considered on an abstract manifold. In differential geometry, there are two main notions of differentiation (of arbitrary tensor fields) that are invariant under coordinate transformations: Lie derivatives, and derivatives with respect to connections. The main difference between these is that taking a derivative with respect to a connection requires an additional geometric structure (e.g. a Riemannian metric or just an abstract connection) on the manifold, but the derivative of a tensor field with respect to a tangent vector is well-defined even if it is not specified how to extend that tangent vector to a vector field; by contrast, when taking a Lie derivative, no additional information about the manifold is needed, but it is impossible to talk about the Lie derivative of a tensor field with respect to a single tangent vector, since the value of the Lie derivative of a tensor field with respect to a vector field X at a point p depends on the value of X in a neighborhood of p, not just at p itself.
References[]
This page uses content from Wikipedia. The original article was at Lie derivative. The list of authors can be seen in the page history. As with the Math Wiki, the text of Wikipedia is available under the Creative Commons Licence. |