I want to get an expression for $\nabla_{a}g_{bc}$

If you are using the Levi-Civita connection, then the covariant derivative of the metric is zero by the definition of that connection.

See this PSE question.

If you don’t know which connection you’re supposed to be using, you’re almost certainly supposed to be using Levi-Cevita’s.

I want to get an expression for $\nabla_{a}g_{bc}$

If you are using the Levi-Civita connection, then the covariant derivative of the metric is zero by the definition of that connection.

See this PSE question.

If you don’t know which connection you’re supposed to be using, you’re almost certainly supposed to be using Levi-Cevita’s. It’s the unique torsion-free connection that preserves lengths and angles under parallel transport.

I want to get an expression for $\nabla_{a}g_{bc}$

If you are using the Levi-Civita connection, then the covariant derivative of the metric is zero by the definition of that connection.

See this PSE question.

I want to get an expression for $\nabla_{a}g_{bc}$

If you are using the Levi-Civita connection, then the covariant derivative of the metric is zero by the definition of that connection.

See this PSE question.

If you don’t know which connection you’re supposed to be using, you’re almost certainly supposed to be using Levi-Cevita’s.