Second covariant derivative

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: [1]

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

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

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

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

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

Thus

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

When the torsion tensor is zero, so that ${\displaystyle [u,v]=\nabla _{u}v-\nabla _{v}u}$, we may use this fact to write Riemann curvature tensor as [2]

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

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

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

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

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

we find

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

This can be rewritten as

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

so we have

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

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