In the math branches of differential geometry and vector calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to another two tangent vector fields.
Formally, given a (pseudo)-Riemannian manifold (M, g) associated with a vector bundle E â M, let â denote the Levi-Civita connection given by the metric g, and denote by ÃÂ(E) the space of the smooth sections of the total space E. Denote by T<sup>*</sup>M the cotangent bundle of M. Then the second covariant derivative can be defined as the composition of the two âÂÂs as follows:
For example, given vector fields u, v, w, a second covariant derivative can be written as
by using abstract index notation. It is also straightforward to verify that
Thus
When the torsion tensor is zero, so that , we may use this fact to write Riemann curvature tensor as
Similarly, one may also obtain the second covariant derivative of a function f as
Again, for the torsion-free Levi-Civita connection, and for any vector fields u and v, when we feed the function f into both sides of
we find
This can be rewritten as
so we have
That is, the value of the second covariant derivative of a function is independent on the order of taking derivatives.