Second covariant derivative explained

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: ^[1]

\Gamma(E) \stackrel{\nabla}{\longrightarrow} \Gamma(T ^{*M} ⊗ E) \stackrel{\nabla}{\longrightarrow} \Gamma(T ^{*M} ⊗ T ^{*M} ⊗ E). For example, given vector fields u, v, w, a second covariant derivative can be written as(\nabla ^{2} _{u,v} 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 ^{2} _{u,v} w) ^{a} + (\nabla _{\nabla _{u} v} w) ^{a.} Thus\nabla ^{2} _{u,v} w = \nabla _{u} \nabla _{v} w - \nabla _{\nabla _{u} v} w. When the torsion tensor is zero, so that [u,v] = \nabla _{uv-\nabla} _{vu} , we may use this fact to write Riemann curvature tensor as ^[2] R(u,v) w = \nabla ^{2} _{u,v} w - \nabla ^{2} _{v,u} w. Similarly, one may also obtain the second covariant derivative of a function f as\nabla ^{2} _{u,v} 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 ^{2} _{u,v} f = \nabla ^{2} _{v,u} f. That is, the value of the second covariant derivative of a function is independent on the order of taking derivatives.

Notes and References

1. Web site: Parker. Thomas H.. Geometry Primer. 2 January 2015., pp. 7
2. Web site: Jean Gallier and Dan Guralnik. Chapter 13: Curvature in Riemannian Manifolds. 2 January 2015.