# Why do we need Lie derivative?

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?

• it does not change if you change the Riemannian metric. – Will Jagy Mar 24 '13 at 20:21
• Disclaimer: barely learned about this stuff, so this is a rough sketch of vague recollections, that hopefully others may flesh out. As was said, the Lie derivative is defined before the connection is added. The other type of derivative (forgot name) can be equal to the Lie derivative if just the right connection is chosen. If a connection is not chosen this way, there is another quantity which measures the difference between the Lie derivative and the other one. I think it was called torsion. – Brady Trainor Mar 24 '13 at 20:33
• You should also note that when one defines the Riemannian Connection on a Riemannian manifold in terms of the Koszul formula, then needs the Lie derivative (just take a look at all of the bracket terms). You might take a look at Peter Peterson's textbook Riemannian Geometry." He makes use of Lie derivatives throughout, introducing them early and often (especially when compared to other texts like Do Carmo or John Lee). – THW Mar 25 '13 at 12:47

## 1 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.

• Sorry, I just realised you didn't ask why Lie derivatives and covariant derivatives are different, you just wanted to know why we care about Lie derivatives. They are just a tool in a toolkit and you don't say we should throw away the mallet just because we have a hammer. Sometimes the Lie derivative is perfectly suited to our needs. – muzzlator Mar 24 '13 at 20:34
• E.g. Frobenius integrability theorem is clearly stated using Lie derivatives. – Elchanan Solomon Mar 24 '13 at 20:41