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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.