Recall that is defined to be

.

Let

.

The way to understand is to look at the following 4-covariant tensor

.

As can be seen, the components of are .

We first obtain the following result.

Theorem 1. The curvature tensor satisfies the following property.

*Proof*.

The proof relies on the definition of the 4-covariant tensor above. To be precise, one has

and

.

Since

then . This comes from the definition of curvature tensor and the fact that

.

Similarly, for the latter case, one can argue as follows

.

We now use the fact that is a metric connection. Indeed,

Thus

Hence

.

The above identity also holds if we replace by a vector field . Thus

which implies

.

Therefore, .

Corollary 1. and .

Theorem 2 (the first Bianchi identity). The curvature tensor satisfies the following property.

*Proof*. Since

then

.

Similarly,

.

Since is torsion free, one gets

.

As a consequence,

.

Now

which implies

.

If we change by , by we then obtain

which implies, by using Theorem 1,

.

Corollary 2. .

Corollary 3. Followed from the proof of Theorem 2, by pairing with to the both sides one has.

Theorem 3. The curvature tensor satisfies the following property.

*Proof*. By the first Bianchi indentity,

which implies

.

Thus

.

Similarly, by changing , , and one gets

.

Hence by using Theorem 1.

Theorem 4 (the second Bianchi identity). The curvature tensor satisfies the following property.

*Proof*. One can use the normal coordinates in order to simplify the calculation. Indeed, normal coordinates tell us at a given point that

and

for all , , . Thus,

which implies

Hence

which implies

.

Similarly, we can write down and . Summing up we get the desired result.

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