Christoffel Symbols - Definition - Ricci Rotation Coefficients (asymmetric Definition)

Ricci Rotation Coefficients (asymmetric Definition)

When we choose the basis orthonormal: then . This implies that

$\omega^i{}_{k\ell}=\frac{1}{2}\eta^{im} \left( c_{mk\ell}+c_{m\ell k} - c_{k\ell m} \right)\,$

and the connection coefficients become antisymmetric in the first two indices:

where .

In this case, the connection coefficients are called the Ricci rotation coefficients.

Equivalently, one can define Ricci rotation coefficients as follows:

where is an orthonormal non holonomic basis and its co-basis.

Read more about this topic: Christoffel Symbols, Definition

“The lazy manage to keep up with the earth’s rotation just as well as the industrious.”
—Mason Cooley (b. 1927)