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.

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# Levi-Civita Symbol and indefinite metrics

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