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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 TM the cotangent bundle of M. Then the second covariant derivative can be defined as the composition of the two ∇s as follows:

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

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 ${\displaystyle =\nabla _{u}v-\nabla _{v}u}$, we may use this fact to write Riemann curvature tensor as

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

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

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

we find

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

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.

Notes

1. Parker, Thomas H. "Geometry Primer" (PDF). Retrieved 2 January 2015., pp. 7
2. Jean Gallier and Dan Guralnik. "Chapter 13: Curvature in Riemannian Manifolds" (PDF). Retrieved 2 January 2015.

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