Second covariant derivative

See also: Exterior covariant derivative

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.

Definition

Formally, given a (pseudo)-Riemannian manifold (M, g) associated with a vector bundle E → M, let ∇ denote the Levi-Civita connection given by the metric g, and denote by Γ(E) the space of the smooth sections of the total space E. Denote by T^*M the cotangent bundle of M. Then the second covariant derivative can be defined as the composition of the two ∇s as follows:

$\Gamma (E){\stackrel {\nabla }{\longrightarrow }}\Gamma (T^{*}M\otimes E){\stackrel {\nabla }{\longrightarrow }}\Gamma (T^{*}M\otimes T^{*}M\otimes E).$

For example, given vector fields u, v, w, a second covariant derivative can be written as

$(\nabla _{u,v}^{2}w)^{a}=u^{c}v^{b}\nabla _{c}\nabla _{b}w^{a}$

by using abstract index notation. It is also straightforward to verify that

$(\nabla _{u}\nabla _{v}w)^{a}=u^{c}\nabla _{c}v^{b}\nabla _{b}w^{a}=u^{c}v^{b}\nabla _{c}\nabla _{b}w^{a}+(u^{c}\nabla _{c}v^{b})\nabla _{b}w^{a}=(\nabla _{u,v}^{2}w)^{a}+(\nabla _{\nabla _{u}v}w)^{a}.$

Thus

$\nabla _{u,v}^{2}w=\nabla _{u}\nabla _{v}w-\nabla _{\nabla _{u}v}w.$

When the torsion tensor is zero, so that $[u,v]=\nabla _{u}v-\nabla _{v}u$, we may use this fact to write Riemann curvature tensor as

$R(u,v)w=\nabla _{u,v}^{2}w-\nabla _{v,u}^{2}w.$

Similarly, one may also obtain the second covariant derivative of a function f as

$\nabla _{u,v}^{2}f=u^{c}v^{b}\nabla _{c}\nabla _{b}f=\nabla _{u}\nabla _{v}f-\nabla _{\nabla _{u}v}f.$

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

$\nabla _{u}v-\nabla _{v}u=[u,v]$

we find

$(\nabla _{u}v-\nabla _{v}u)(f)=[u,v](f)=u(v(f))-v(u(f)).$.

This can be rewritten as

$\nabla _{\nabla _{u}v}f-\nabla _{\nabla _{v}u}f=\nabla _{u}\nabla _{v}f-\nabla _{v}\nabla _{u}f,$

so we have

$\nabla _{u,v}^{2}f=\nabla _{v,u}^{2}f.$

That is, the value of the second covariant derivative of a function is independent on the order of taking derivatives.