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In the last section we found that in a local inertial frame the Riemann tensor could be written as

Differentiating with respect to we get

From this equation, the symmetry and the fact that partial derivatives commute, it is easy to show that

This equation is valid in a local inertial frame, therefore in a general frame we get

This is a tensor equation, therefore valid in any coordinate system. It is called the Bianchi identities , and will be very important for our work.