Unlike the case of partial differentiation for functions, in general, for sections of a vector bundle, partial differentiations for different directions do not commute. The failure of the commutativity is measured by the curvature of the connection. The commutativity of partial differentiation of a function in two different directions is equivalent of the composite of two exterior differentiations applied to functions. We do this for a real manifold of real dimension and a smooth vector bundle of rank over which can be a complex vector bundle or a real one. Take a local smooth basis of . We apply twice to and get a local section of , where is the dual of the tangent bundle of . By skew-symmmetrizing in the two arguments for the tangent vectors of , we get a local -valued -form . We can express in terms of the connection and its exteiror derivative as follows. We use the column vector with components .

We define the curvature to be and denote it by . It is an -valued -form on , because if is a local smooth matrix-valued function, then

Let us look at the curvature tensor is local coordinates. With respect to a local frame , we have

Using the Christoffel symbol of the connection , we have

and

The curvature tensor satisfies a Bianchi identity. The Bianchi identity is

In local coordinates, this says that

We would like to remark that two partial covariant differentiations in general do not commute, and the noncommutativity gives rise to the curvature. However, the skew-symmetrization of the results of three partial covariant differentiations becomes zero, which is the Bianchi identity. We can also define the curvature in invariant formulation. The curvature is defined to measure the failure of the commutativity of partial covariant differentiation. So for tangent vector fields , and a smooth local section of , we should consider . However, when and are arbitrary tangent vector fields instead of tangent vector fields defined by coordinate functions, we do not expect to have commutativity of differentiations along and even in the case of functions, because for a smooth function , in general, is nonzero and is equal to the derivative of along the direction of the Lie bracket of and . So we have to make accommodation for that and consider

For smooth functions , , , we have

because of the following computations.

So the value of at a point depends only on the values of , , at the point . For

it follows from definitions that

So the curvature can be defined in an invariant way by

Let us consider the case when is a complex manifold of complex dimension and is a holomorphic vector bundle. Suppose we have a Hermitian metric along the fibers of and the connection is a complex metric connection. When we choose a local holomorphic basis , we have

This shows that is actually an -valued -form on , and it simply equals when expressed in terms of a local holomorphic basis of . The Bianchi identity for takes on a simpler form. In terms of local coordinates, the Bianchi identity is

Since the curvature tensor is of type , we simply have

Analogously, we have also

We denote by the trace of so that is a -form on . The -form is the curvature form of the determinant bundle of with the metric induced from the metric of , because with respect to a local holomorphic basis,

and

Now we come back to the real case. Consider the real manifold of real dimension and the tangent vector bundle with a Riemannian metric. Then the curvature tensor is given by

In this case, we have another Bianchi identity. The earlier one is usually referred to as the second Bianchi identity. The earlier one is usually referred to as the second Bianchi identity, and the one we are going to discuss is called the first Bianchi identity. At one point, choose a coordinate system so all vanish at that point. Then at the point, we simply have

from which it follows that

because of the symmetry of in and . This is the first Bianchi identity. The first Bianchi identity is a consequence of the torsion-free condition. We can see this more clearly in the following way. Suppose is its dual frame. Let with

be the matrix valued -form of the connection. The torsion-free condition is

Taking exterior derivative of both sides and using the original equation to get rid of , we get

When we write

we have

which is the first Bianchi identity. There is another symmetry we want. Let

From

we have

Thus,

We consider now the Kähler case. From

and the fact that the only nonvanishing components of the Christoffel symbols are of type and , we conclude that when and are of different type, must vanish. So vanishes unless and are of different type. We can also see this from the fact that the curvature of the complex metric connection is of type . By the above symmetry, we conclude that the only nonzero components of the curvature tensor are of type , , , .