Opposite of the matrix of the metric tensor

Being given a Frame of reference, the matrix of the metric Tenseur in Composantes contravariantes $g^\; \{ij\}$ is the opposite matrix of the matrix of the metric tensor in Composantes covariantes:

$g^\; \{ij\}\; g\_\; \{jk\}\; =\; \backslash \; delta^i\_k$. In other words, the metric tensor is its clean opposite.

Demonstration

One defined (cf Transformation contraco) the metric tensor reverses $(g^\; \{-\; 1\})\; ^\; \{ij\}$ like the reverse of $g\_\; \{ij\}$. While utilizing twice the metric tensor, one obtains his expression covariante

$\backslash \; bigl\; (g^\; \{-\; 1\}\; \backslash \; bigr)\; \_\; \{ij\}\; =\; g\_\; \{ik\}\; g\_\; \{jl\}\; \backslash \; bigl\; (g^\; \{-\; 1\}\; \backslash \; bigr)\; ^\; \{kl\}$

g_ {ik} g_ {jl} \ bigl (g^ {- 1} \ bigr) ^ {lk}

g_{ik} \delta^k_j g_{ij}.

The metric tensor is thus its own reverse, c.q.f.d.

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