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## Summary:

- Deriving the expression for the covariant derivative of a covector field in components

## Main Question or Discussion Point

I am trying to derive the expression in components for the covariant derivative of a covector (a 1-form), i.e the Connection symbols for covectors.

What people usually do is

- https://www.physicsforums.com/threads/how-do-i-derive-the-covariant-derivative-for-covectors-lower-index.689141/

- https://math.stackexchange.com/questions/1069916/covariant-derivative-for-a-covector-field

- https://math.stackexchange.com/questions/1499513/covariant-derivative-of-a-covariant-vector

Well, I find something disturbing: the covariant derivative doesn't have such a product rule between a covector field and a vector field, or even between two vector fields. The only product rule it has is the Leibniz rule (which is one of the defining property of the axiomatic definition of the covariant derivative) between a vector field and a scalar field (a function on the manifold), but definitely no such thing between a vector and a covector.

What is an explanation for this, and a more rigorous derivation?

What people usually do is

- take the covariant derivative of the covector acting on a vector, the result being a scalar
- Invoke a product rule to develop in "(covariant derivative of vector)(covector) + (covariant derivative of covector)(vector) = Covariant derivative of scalar"
- substract the covariant derivative of the vector which is already known, leaving the expression for the covariant derivative for the covector on the other side.

- https://www.physicsforums.com/threads/how-do-i-derive-the-covariant-derivative-for-covectors-lower-index.689141/

- https://math.stackexchange.com/questions/1069916/covariant-derivative-for-a-covector-field

- https://math.stackexchange.com/questions/1499513/covariant-derivative-of-a-covariant-vector

Well, I find something disturbing: the covariant derivative doesn't have such a product rule between a covector field and a vector field, or even between two vector fields. The only product rule it has is the Leibniz rule (which is one of the defining property of the axiomatic definition of the covariant derivative) between a vector field and a scalar field (a function on the manifold), but definitely no such thing between a vector and a covector.

What is an explanation for this, and a more rigorous derivation?