Second covariant derivative - Alchetron, the free social encyclopedia

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:

Γ ( E ) ⟶ ∇ Γ ( T ∗ M ⊗ E ) ⟶ ∇ Γ ( T ∗ M ⊗ T ∗ M ⊗ E ) .

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

( ∇ u , v 2 w ) a = u c v b ∇ c ∇ b w a

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

( ∇ u ∇ v w ) a = u c ∇ c v b ∇ b w a = u c v b ∇ c ∇ b w a + ( u c ∇ c v b ) ∇ b w a = ( ∇ u , v 2 w ) a + ( ∇ ∇ u v w ) a .

Thus

∇ u , v 2 w = ∇ u ∇ v w − ∇ ∇ u v w .

One may use this fact to write Riemann curvature tensor as follows:

R ( u , v ) w = ∇ u , v 2 w − ∇ v , u 2 w .

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

∇ u , v 2 f = u c v b ∇ c ∇ b f = ∇ u ∇ v f − ∇ ∇ u v f .

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

∇ u v − ∇ v u = [ u , v ] .

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

( ∇ u v − ∇ v u ) ( f ) = [ u , v ] ( f ) = u ( v ( f ) ) − v ( u ( f ) ) . ∇ ∇ u v f − ∇ ∇ v u f = ∇ u ∇ v f − ∇ v ∇ u f .

Thus

∇ u , v 2 f = ∇ v , u 2 f .

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

References

Second covariant derivative Wikipedia

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