Levi-Civita connection and Christoffel symbols

In summary, the conversation discusses how to show that g(dσk, σp ∧ σq) = Γipq - Γiqp. The solution involves using the definition of the connection form and the metric compatibility. The justification for saying that dσk = ωjiσj also seems valid.
  • #1
WendysRules
37
3

Homework Statement


Show that [tex]g(d \sigma ^k, \sigma _p \wedge \sigma _q) = \Gamma _{ipq} - \Gamma _{iqp}[/tex]

Homework Equations


Given $$\omega_{ij}=\hat e_i \cdot d \hat e _j = \Gamma_{ijk} \sigma^k$$, we can also say that $$d \hat e_j = \omega^i_j \hat e_i$$. Where $$\sigma^k, \sigma_p, \sigma_q$$ are to be one forms.

The Attempt at a Solution


I'm self -studying this material, and I just want to make sure I'm making the right connections!

So what I did was to say, $$g(d\sigma^k, \sigma_p \wedge \sigma_q) =g(\omega_{ji} \sigma^j, \sigma_p \wedge \sigma_q) = g(\Gamma_{jik} \sigma^k \wedge \sigma^j, \sigma_p \wedge \sigma_q) = g(-\Gamma_{ijk} \sigma^k \wedge \sigma^j, \sigma_p \wedge \sigma_q)$$

$$= -\Gamma_{ijk}g(\sigma^k \wedge \sigma^j, \sigma_p \wedge \sigma_q) = -\Gamma_{ijk}[\delta^k_p \delta^j_q - \delta^k_q \delta^j_p] = -\Gamma_{ijk}\delta^k_p \delta^j_q + \Gamma_{ijk} \delta^k_q \delta^j_p = -\Gamma_{iqp}+\Gamma_{ipq} = \Gamma_{ipq} - \Gamma_{iqp}$$

Cool, that's what we wanted, but I'm not sure if everything I did was valid, so I'm just seeing if it's fine what I did.
My justification for saying $$d\sigma^k = \omega_{ji} \sigma^j $$ is that we see that $$d \hat e_j = \omega^i_j \hat e_i$$ since $$\hat e_j $$is a basis vector, and $$\sigma_i$$ is a one form that can also form a basis, they should have the same connection. Thus, $$d\sigma_i = \omega^k_i \sigma_k = \omega^k_i g_{kj} \sigma^j = \omega_{ji} \sigma^j$$

We know $$\Gamma_{jik} = - \Gamma_{ijk}$$ by metric compatibility.

Sorry for the long post, just need some pointers, or reassurance that this is all right! (I've posted this on another forum, but no replies).
 
Physics news on Phys.org
  • #2

Thank you for your post. Your solution looks correct to me. You have correctly used the definition of the connection form and the metric compatibility to arrive at the desired result. Your justification for saying that $d\sigma^k = \omega_{ji} \sigma^j$ also seems valid. Keep up the good work, and don't hesitate to ask if you have any further questions.
 

Related to Levi-Civita connection and Christoffel symbols

1. What is the Levi-Civita connection?

The Levi-Civita connection is a mathematical tool used in the study of differential geometry. It is a way to define a unique connection on a smooth manifold that is compatible with the metric structure of the manifold. It allows for the calculation of covariant derivatives and geodesics, which are essential in understanding the geometry of a manifold.

2. What are Christoffel symbols?

Christoffel symbols are a set of numbers that represent the components of the Levi-Civita connection in a specific coordinate system. They are named after the mathematician Elwin Bruno Christoffel and are used to calculate covariant derivatives and geodesics in a particular coordinate system.

3. How are Levi-Civita connection and Christoffel symbols related?

The Levi-Civita connection is a mathematical concept that can be represented by Christoffel symbols in a specific coordinate system. The Christoffel symbols are used to calculate the components of the Levi-Civita connection and, in turn, the covariant derivatives and geodesics on a manifold.

4. What is the significance of the Levi-Civita connection and Christoffel symbols?

The Levi-Civita connection and Christoffel symbols are essential in the study of differential geometry and the understanding of the geometry of a manifold. They allow for the calculation of covariant derivatives and geodesics, which are crucial in describing the curvature and properties of a manifold.

5. How are Levi-Civita connection and Christoffel symbols used in physics?

The concepts of Levi-Civita connection and Christoffel symbols are used extensively in physics, particularly in the fields of general relativity and electromagnetism. In general relativity, they are used to describe the curvature of spacetime and the motion of particles in a gravitational field. In electromagnetism, they are used to calculate the electromagnetic field strength tensor and the Lorentz force on a charged particle.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
20
Views
2K
  • Differential Geometry
Replies
5
Views
3K
  • MATLAB, Maple, Mathematica, LaTeX
Replies
6
Views
6K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
Replies
9
Views
1K
  • Special and General Relativity
Replies
4
Views
717
  • Differential Geometry
Replies
5
Views
6K
  • Math Proof Training and Practice
3
Replies
100
Views
7K
  • Special and General Relativity
Replies
7
Views
6K
Back
Top