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.

Definition

Formally, given a (pseudo)-Riemannian manifold (M, g) associated with a vector bundle EM, 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]

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

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

Thus

When the torsion tensor is zero, so that , we may use this fact to write Riemann curvature tensor as [2]

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

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

we find

.

This can be rewritten as

so we have

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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