Relation between covariant differential and covariant derivative

In summary, the covariant differential and the covariant derivative are two tensors that are closely related in their purpose and notation. The covariant differential represents the infinitesimal change of a vector field along a curve, while the covariant derivative represents the rate of change of a vector field along a given direction. They are used to study the curvature and geometry of a Riemannian manifold, but can also be defined and used in other types of manifolds. The notation for covariant differential is $dV$ and for covariant derivative is $\nabla V$.
  • #1
center o bass
560
2
In Theodore Frankel's book, "The Geometry of Physics", he observes at page 248 that the covariant derivative of a vector field can be written as

$$\nabla_X v = e_iX^j (v^i_{,j} + \omega^i_{jk} v^k)= e_i(dv^i(X) + \omega^i_k(X) v^k) = e_i (dv^i + \omega^i_k v^k)(X)$$

where ##\omega^i_k = \omega^i_{jk} dx^j##, such that we can write

##\nabla v = e_i\otimes (dv^i + \omega^i_k v^k)##. He then goes on to define the "covariant differential" of a vector valued p-form ##\alpha = e_i \otimes \alpha^i## as

$$ \nabla \alpha = e_i \otimes (d\alpha^i + \omega^i_k \wedge \alpha^k)$$

I was then left wondering what relation the "covariant differential" had to the covariant derivative of the vector valued p-form? Since we had that ##\nabla v (X) = \nabla_X v## for a 'vector valued zero form', does one have something like ##\nabla \alpha (X) = \nabla_X \alpha## for a vector valued p-form? Is there any relation here?
 
Physics news on Phys.org
  • #2

Thank you for bringing up this interesting observation from Theodore Frankel's book. The relation between the "covariant differential" of a vector valued p-form and the covariant derivative of the same form is indeed an important one in differential geometry.

To understand this relation, we must first understand the concept of a covariant derivative. The covariant derivative is a generalization of the usual derivative in calculus, which takes into account the curvature of a space. In the context of differential geometry, the covariant derivative of a vector field measures how the vector field changes as we move along a curve in the space. It is defined using the connection on the space, which is represented by the ##\omega^i_k## in the equation you mentioned.

Now, in the equation for the covariant derivative of a vector field, we see that the connection ##\omega^i_k## appears in two places - as a coefficient for the derivative of the vector field, and as a coefficient for the vector field itself. This is because the covariant derivative not only measures the change of a vector field along a curve, but also takes into account how the basis vectors of the space change as we move along the curve. This is where the covariant differential comes in.

The covariant differential of a vector valued p-form, as defined by Frankel, takes into account both the change of the form itself and the change of the basis vectors of the space. This is why we see the wedge product ##\wedge## in the equation, which represents the change of basis vectors. So, in essence, the covariant differential is a generalization of the usual differential in calculus, just like the covariant derivative is a generalization of the usual derivative.

Now, to answer your question, the relation between the "covariant differential" and the covariant derivative is that they are essentially the same concept, just applied to different objects. The notation may be different, but the underlying idea is the same. So, we do have ##\nabla \alpha (X) = \nabla_X \alpha## for a vector valued p-form, just like we have ##\nabla v (X) = \nabla_X v## for a vector valued zero form.

I hope this helps to clarify the relation between the "covariant differential" and the covariant derivative. If you have any further questions, please do not hesitate to ask.

Best
 

1. What is the difference between covariant differential and covariant derivative?

The covariant differential is a tensor that represents the infinitesimal change in a vector field along a curve, while the covariant derivative is a tensor that represents the rate of change of a vector field along a given direction.

2. How are covariant differential and covariant derivative related?

The covariant differential is used to define the covariant derivative by taking the limit of the covariant differential as the length of the curve approaches zero. In other words, the covariant derivative is the derivative of a vector field along a given direction.

3. What is the purpose of using covariant differential and covariant derivative?

Covariant differential and covariant derivative are used to study the curvature and geometry of a Riemannian manifold. They allow us to define and calculate the change of a vector field along a given direction, which is essential in understanding the behavior of objects in curved spaces.

4. What is the notation for covariant differential and covariant derivative?

The notation for covariant differential is $dV$, where $V$ is the vector field. The notation for covariant derivative is $\nabla V$, where $\nabla$ is the covariant derivative operator and $V$ is the vector field.

5. Are covariant differential and covariant derivative only applicable in Riemannian manifolds?

No, covariant differential and covariant derivative can also be defined and used in other types of manifolds, such as Lorentzian manifolds. However, their specific properties and applications may differ in different types of manifolds.

Similar threads

  • Differential Geometry
Replies
5
Views
2K
  • Differential Geometry
Replies
6
Views
2K
  • Differential Geometry
Replies
15
Views
4K
  • Differential Geometry
Replies
7
Views
3K
  • Differential Geometry
Replies
14
Views
3K
Replies
11
Views
2K
  • Advanced Physics Homework Help
Replies
5
Views
2K
  • Differential Geometry
2
Replies
42
Views
11K
  • Differential Geometry
Replies
4
Views
4K
  • Differential Geometry
Replies
8
Views
3K
Back
Top