If a manifold is equipped with Levi-Civita connection, Why do we need Lie derivative?
In Euclidean space to calculate directional derivative of a vector field V along W, we parallel transport V along W and … (same as what did in Riemannian manifolds by covariant derivative). In other words, I think that Generalized directional derivative in Euclidean Space is covariant derivative in Riemannian manifolds.
Can anyone help me to understand necessity of existence of Lie derivative?
Best Answer
Lie derivatives don't use the connection at all. They operate on the notion of evaluating a vector field along an integral curve of another vector field, this is inherently different to the notion of parallel transport.
Look at what happens when you take the commutator of integral curves, you get the Lie derivative. On the other hand if you take the commutator of parallel transport, you get the curvature tensor.