Lecture 11: Symmetry (International Winter School on Gravity and Light 2015)

[@ivanphi]

I am confused with the way the Lie Derivative is introduced as a limit. Which is the metric (in the topological sense that allows us to define open balls etc...) on the space of (0,2)-tensor fields that allows us to make sense out of such a limit?

[@vs-cw1wc]

I know it is late but this is actually a point that needs to be clarified (You have a very mathematical mind...). As you pointed out the expression inside the limit symbol is just a (0,2)-tensor field on the manifold, which is defined on every point of the manifold as a linear map from 2 copies of the tangent space to the real numbers. So the easiest way to make sense of this limit is to say that the tensor field tends to zero if at each point the linear map (as described above) tends to the zero map. The limit of linear maps between finite dimensional vector spaces is easy to define using the standard topology of those vector spaces. It is also possible, I guess, to provide a topology to Gamma(T*^2M) but I do not see the need here.