Levi-Civita Symbol and indefinite metrics

In summary, the Levi-Civita symbol is defined on a pseudo-Riemannian manifold and can be used to compute the determinant of a tangent space endomorphism. There are various identities for the symbol, but a specific reference for computing them is not readily available.
  • #1
Geometry_dude
112
20
Let ##(M,g)## be an ##n##-dimensional pseudo-Riemannian manifold of signature ##(n_+, n_-)## and define the Levi-Civita symbol via
$$\varepsilon_{i_1 \dots i_n} \, \theta^{i_1 \dots i_n} = n! \, \theta^{[1 \dots n]} =
\theta^1 \wedge \dots \wedge \theta^n$$
where ##\theta^1, \dots, \theta^n## is a basis of covectors and the brackets denote the antisymmetrization operation.
With this definition I was able to prove the following formula for the determinant of a tangent space endomorphism (i.e. a "matrix") ##A##
$$\det A = n! \, A^{1}{}_{[1} \cdots A^{n}{}_{n]}.$$
My first question is: How do I compute
$$\varepsilon_{i_1 \dots i_n} \varepsilon^{i_1 \dots i_n} \equiv
\varepsilon_{i_1 \dots i_n} \, g^{i_1j_1} \cdots g^{i_n j_n}\, \varepsilon_{j_1 \dots j_n}$$
in a formal manner?
How do I compute the other common identities for the Levi-Civita Symbol like
$$\varepsilon_{i_1 \dots i_k j_{1} \dots j_{n-k}} \varepsilon^{j_{1} \dots j_{n-k} l_1 \dots l_k} = \, ? $$

I have browsed loads of differential geometry books, but none do this seemingly basic thing explicitly.
 
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  • #2
Noone has any idea? Actually, I was trying to prove this to understand the Hodge operator.

Does anyone have a good reference?

The Levi-Civita symbol as defined here is just the standard one:

$$\varepsilon_{i_1 \dots i_n} = n! \, \delta^{[1}_{i_1} \cdots \delta^{n]}_{i_n}$$
 
  • #3
Have you seen the determinant expressions in http://en.wikipedia.org/wiki/Levi-Civita_symbol ?
Are you specifically looking for something that uses metrics with general signatures (so there may be factors of (-1) for each timelike direction)?

Look at the google book page from this google search: de felice alternating tensors
 
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1. What is the Levi-Civita symbol and how is it used in mathematics?

The Levi-Civita symbol, also known as the permutation symbol, is a mathematical symbol used to represent the sign of a permutation of a set of numbers. It is used in various branches of mathematics, such as vector calculus and differential geometry, to simplify calculations involving determinants and cross products.

2. What is the significance of the Levi-Civita symbol in indefinite metrics?

In indefinite metrics, the Levi-Civita symbol plays a crucial role in the formulation of the metric tensor. The metric tensor is used to define the distance between two points in a curved space, and the Levi-Civita symbol helps to account for the non-commutativity of partial derivatives in these spaces.

3. Can the Levi-Civita symbol be generalized to higher dimensions?

Yes, the Levi-Civita symbol can be generalized to higher dimensions. In three-dimensional space it is represented as a 3x3 matrix, but in higher dimensions it can be represented as a higher-order tensor with more indices.

4. What is the relation between the Levi-Civita symbol and cross products?

The Levi-Civita symbol is often used to simplify cross product calculations in three-dimensional space. The cross product of two vectors can be written as a determinant involving the Levi-Civita symbol, making it easier to calculate.

5. Are there any applications of the Levi-Civita symbol outside of mathematics?

Yes, the Levi-Civita symbol has applications in physics and engineering as well. It is used in the study of electromagnetism, fluid mechanics, and special relativity, among others. It also has applications in computer graphics and image processing algorithms.

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