then may be written as In this local frame, the connection form becomes with being the Christoffel symbol; again, the index i runs over (the dimension of the underlying manifold M) while j and k run over , the dimension of the fiber. Inserting and turning the crank, one obtains where now identifiable as the Riemann curvature tensor. This is written in the style commonly employed in many textbooks on general relativity from the middle-20th century (with several notable exceptions, such as MTW, that pushed early on for an index-free notation). Again, the indices i and j run over the dimensions of the manifold M, while r and k run over the dimension of the fibers. Tangent-bundle style The above can be back-ported to the vector-field style, by writing as the standard basis elements for the tangent bundle TM. One then defines the curvature tensor as so that the spatial directions are re-absorbed, resulting in the notation Alternately, the spatial directions can be made manifest, while hiding the indices, by writing the expressions in terms of vector fields X and Y on TM. In the standard basis, X is and likewise for Y. After a bit of plug and chug, one obtains where is the Lie derivative of the vector field Y with respect to X. To recap, the curvature tensor maps fibers to fibers: so that To be very clear, are alternative notations for the same thing. Observe that none of the above manipulations ever actually required the bundle metric to go through. One can also demonstrate the second Bianchi identity without having to make any use of the bundle metric. Yang–Mills connection The above development of the curvature tensor did not make any appeals to the bundle metric. That is, they did not need to assume that D or A were metric connections: simply having a connection on a vector bundle is sufficient to obtain the above forms. All of the different notational variants follow directly only from consideration of the endomorphisms of the fibers of the bundle. The bundle metric is required to define the Hodge star and the Hodge dual; that is needed, in turn, to define the Laplacian, and to demonstrate that Any connection that satisfies this identity is referred to as a Yang–Mills connection. It can be shown that this connection is a critical point of the Euler–Lagrange equations applied to the Yang–Mills action