Let \(M\) be a smooth manifold and \(F\in \Gamma(T^{(k,l)}TM)\) a smooth tensor field . Then applying the total covariant derivative on \(F\) two times, we obtain a smooth \((k,l+2)\)-tensor field \(\nabla^2F=\nabla(\nabla F)\). We write \(\nabla^2_{X,Y}F(\ldots)=\nabla^2F(\ldots,Y,X)\). Note, that the convention is, to apply \(Y\) first and \(X\) afterwards.

Remark
  • \(\nabla^2_{X,Y}F\) is not the same as \(\nabla_{X}( \nabla_{Y}F)\), it is rather \begin{equation*} \nabla_{X,Y}^2F=\nabla_X(\nabla_YF)-\nabla_{(\nabla_X Y)}F. \end{equation*} [1, Proposition 4.21]
  • Also \(\nabla^2_{X,Y}F\neq \nabla^2_{Y,X}F\) in general (see Ricci identity ).

See also Link to heading

References Link to heading

  1. J. Lee, Introduction to Riemannian Manifolds. Cham: Springer International Publishing, 2018. doi:10.1007/978-3-319-91755-9