Suppose ∇ is a derivative operator on the manifold M. Then there is a (unique) smooth tensor field Rabcd on M such that for all smooth fields ξb,
Rabcd ξb = −2∇[c∇d] ξa —– (1)
Uniqueness is immediate since any two fields that satisfied this condition would agree in their action on all vectors ξb at all points. For existence, we introduce a field Rabcd and do so in such a way that it is clear that it satisfies the required condition. Let p be any point in M and let ξ’b be any vector at p. We define Rabcd ξ’b by considering any smooth field ξb on M that assumes the value ξ’b at p and setting Rabcdξ’b = −2∇[c∇d]ξa. It suffices to verify that the choice of the field ξb plays no role. For this it suffices to show that if ηb is a smooth field on M that vanishes at p, then necessarily ∇[c∇d] ηb vanishes at p as well. (For then we can apply this result, taking ηb to be the difference between any two candidates for ξb.)
The usual argument works. Let λa be any smooth field on M. Then we have,
0 = ∇[c∇d] (ηaλa) = ∇[c ηa ∇d] λa + ηa[c∇d] λa + (∇[c λ|a|) (∇d] ηa) + λa ∇[c∇d] ηa —– (2)
It is to be noted that in the third term of the final sum the vertical lines around the index indicate that it is not to be included in the anti-symmetrization. Now the first and third terms in that sum cancel each other. And the second vanishes at p. So we have 0= λa∇[c ∇d]ηa at p. But the field λa can be chosen so that it assumes any particular value at p. So ∇[c ∇d] ηa = 0 at p.
Rabcd is called the Riemann curvature tensor field (associated with ∇). It codes information about the degree to which the operators ∇c and ∇d fail to commute.