Let ##(M,g)## be an ##n##-dimensional pseudo-Riemannian manifold of signature ##(n_+, n_-)## and define the Levi-Civita symbol via(adsbygoogle = window.adsbygoogle || []).push({});

$$\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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Levi-Civita Symbol and indefinite metrics

**Physics Forums | Science Articles, Homework Help, Discussion**