Lie derivative versus covariant derivative

[RedX]

When calculating the derivative of a vector field X at a point p of a smooth manifold M, one uses the Lie derivative, which gives the derivative of X in the direction of another vector field Y at the same point p of the manifold.

If the manifold is a Riemannian manifold (that is, equipped with a metric tensor), then there is a natural connection called the Levi-Civita connection that also tells you the derivative of a vector field X at a point p of a smooth manifold.

Do these two methods of calculating the derivative give the same result?

And why is the Lie derivative so complicated? It seems the reason is that you need to define a diffeomorphism f: M --> M (an active transformation) because doing so will induce a transformation between tangent spaces: f*: T(M) --> T(M), and to compare 2 vectors at different points you first need to bring one vector to the other through f* so that you can subtract them. This diffeomorphism f is provided by a "flow", which is induced by Y. So what is so special about a diffeomorphism generated by a vector field Y? Why not just use any diffeomorphism to define the derivative, and not necessarily one generated by a vector field?

Also, is there an easy way to see that a flow generated by a vector field is a diffeomorphism?

 

[zhentil]

The Lie derivative is not linear over functions in the first argument, thus it can't be a connection of any kind.

For the flow question, it generates a local one-parameter group of diffeomorphisms (this is existence and uniqueness of linear ODEs). There's no reason to expect that it gives a global diffeomorphism (to see this, punch a hole in the real line and point your arrows at it).

 

[RedX]

For the flow question, it generates a local one-parameter group of diffeomorphisms (this is existence and uniqueness of linear ODEs). There's no reason to expect that it gives a global diffeomorphism (to see this, punch a hole in the real line and point your arrows at it).

Maybe I'm reading my book wrong, but it claims that a flow is a group of global diffeomorphisms.

The flow $$\sigma(t,x)$$, where t is the group parameter and x is a point on the manifold, is given by:

$$\sigma^\mu(t,x)=e^{tX^\mu(x)}x^\mu$$

where $$X^\mu(x)$$ is the vector field at the point x in the $$\mu$$ direction.

Is this equation a diffeomorphism? It seems to be just f(x)=e^g(x)x, which seems like a diffeomorphism.

 

[zhentil]

Maybe I'm reading my book wrong, but it claims that a flow is a group of global diffeomorphisms.

The flow $$\sigma(t,x)$$, where t is the group parameter and x is a point on the manifold, is given by:

$$\sigma^\mu(t,x)=e^{tX^\mu(x)}x^\mu$$

where $$X^\mu(x)$$ is the vector field at the point x in the $$\mu$$ direction.

Is this equation a diffeomorphism? It seems to be just f(x)=e^g(x)x, which seems like a diffeomorphism.

If your book is using coordinates, it certainly seems local.

 

[blue_raver22]

The Lie derivative and covariant derivative are two different ways of calculating the derivative of a vector field on a smooth manifold. While they may seem similar, they have distinct mathematical definitions and applications.

The Lie derivative, also known as the "infinitesimal" or "directional" derivative, is a way of calculating the change of a vector field in the direction of another vector field at a specific point on the manifold. This is useful for studying how a vector field changes along a specific path or trajectory on the manifold.

On the other hand, the covariant derivative is a more general concept that takes into account the curvature of the manifold. It is defined using a connection, such as the Levi-Civita connection, which takes into account the geometric properties of the manifold. This makes it a more versatile and powerful tool for studying the behavior of vector fields on curved spaces.

While the Lie derivative may seem more complicated due to the use of diffeomorphisms and flows, it is necessary in order to compare vectors at different points on the manifold. This is because the tangent spaces at different points are not directly comparable, and the diffeomorphism allows us to "transport" the vector from one point to another in a meaningful way.

The use of a vector field to generate the diffeomorphism is special because it allows for a smooth and continuous transformation between tangent spaces, preserving the geometric structure of the manifold. This is not necessarily the case for any arbitrary diffeomorphism.

As for an easy way to see that a flow generated by a vector field is a diffeomorphism, one can use the fact that the flow preserves the structure of the manifold, such as the metric tensor and the connection. This can be shown through the use of coordinate transformations and the definition of a diffeomorphism.

In summary, while the Lie derivative and covariant derivative may seem similar at first glance, they have distinct definitions and applications in studying vector fields on smooth manifolds. The Lie derivative is useful for studying changes along a specific path, while the covariant derivative takes into account the curvature of the manifold. The use of a vector field to generate a diffeomorphism is necessary for comparing vectors at different points, and the flow generated by the vector field is a special type of diffeomorphism that preserves the geometric structure of the manifold.