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:

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

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

$(\underset{u,v}{\overset{2}{\mathrm{\nabla }}}w{)}^a=u^c{v}^b{\mathrm{\nabla }}_c{\mathrm{\nabla }}_b{w}^a$

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

$({\mathrm{\nabla }}_u{\mathrm{\nabla }}_{v}w{)}^a=u^c{\mathrm{\nabla }}_c{v}^b{\mathrm{\nabla }}_b{w}^a=u^c{v}^b{\mathrm{\nabla }}_c{\mathrm{\nabla }}_b{w}^a+(u^c{\mathrm{\nabla }}_c{v}^b){\mathrm{\nabla }}_b{w}^a=(\underset{u,v}{\overset{2}{\mathrm{\nabla }}}w{)}^a+({\mathrm{\nabla }}_{{\mathrm{\nabla }}_{u}v}w{)}^a.$

Thus

$\underset{u,v}{\overset{2}{\mathrm{\nabla }}}w={\mathrm{\nabla }}_u{\mathrm{\nabla }}_{v}w-{\mathrm{\nabla }}_{{\mathrm{\nabla }}_{u}v}w.$

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

$R(u,v)w=\underset{u,v}{\overset{2}{\mathrm{\nabla }}}w-\underset{v,u}{\overset{2}{\mathrm{\nabla }}}w.$

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

$\underset{u,v}{\overset{2}{\mathrm{\nabla }}}f=u^c{v}^b{\mathrm{\nabla }}_c{\mathrm{\nabla }}_{b}f={\mathrm{\nabla }}_u{\mathrm{\nabla }}_{v}f-{\mathrm{\nabla }}_{{\mathrm{\nabla }}_{u}v}f.$

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

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

we find

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

This can be rewritten as

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

so we have

$\underset{u,v}{\overset{2}{\mathrm{\nabla }}}f=\underset{v,u}{\overset{2}{\mathrm{\nabla }}}f.$

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