Suppose ∇ is a derivative operator on the manifold M. Then there is a (unique) smooth tensor field R^{a}_{bcd} on M such that for all smooth fields ξ^{b},

R^{a}_{bcd} ξ^{b} = −2∇[_{c}∇_{d}] ξ^{a} —– (1)

Uniqueness is immediate since any two fields that satisfied this condition would agree in their action on all vectors ξ^{b} at all points. For existence, we introduce a field R^{a}_{bcd} 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 R^{a}_{bcd} ξ’^{b} by considering any smooth field ξ^{b} on M that assumes the value ξ’^{b} at p and setting R^{a}_{bcd}ξ’^{b} = −2∇[_{c}∇_{d}]ξ^{a}. It suffices to verify that the choice of the field ξ^{b} plays no role. For this it suffices to show that if η^{b} is a smooth field on M that vanishes at p, then necessarily ∇[_{c}∇_{d}] η^{b} vanishes at p as well. (For then we can apply this result, taking η^{b} to be the difference between any two candidates for ξ^{b}.)

The usual argument works. Let λ_{a} be any smooth field on M. Then we have,

0 = ∇[_{c}∇_{d}] (η^{a}λ_{a}) = ∇[_{c} η^{a} ∇_{d}] λ_{a} + η^{a}[_{c}∇_{d}] λ_{a} + (∇[_{c} λ_{|a|}) (∇_{d}] η^{a}) + λ_{a} ∇[_{c}∇_{d}] η^{a} —– (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= λ_{a}∇[_{c} ∇_{d}]η^{a} at p. But the field λ_{a} can be chosen so that it assumes any particular value at p. So ∇[_{c} ∇_{d}] η^{a} = 0 at p.

R^{a}_{bcd} 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.