Second 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. 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]

\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.

One may use this fact to write Riemann curvature tensor as follows: ^[2]

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.

Since Levi-Civita connection is torsion-free, for any vector fields u and v, we have

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

By feeding the function f on both sides of the above equation, we have

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

\nabla _{{\nabla _{u}v}}f-\nabla _{{\nabla _{v}u}}f=\nabla _{u}\nabla _{v}f-\nabla _{v}\nabla _{u}f.

Thus

\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

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

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