Suvarna Garge

Second covariant derivative

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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. 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:

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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