I am reading Harmonic Mappings of Riemannian Manifolds by Eells and Sampson. In chapter 2, the author(s) used the divergence theorem, which does not look like the usual divergence theorem for manifolds, and I can't figure out why it is correct.

Simply speaking, in this paper, a family of quantities $\xi^j\ (j=1,\cdots,m)$ is defined using local coordinates $(x^1,\cdots,x^m)$ on a Riemannian manifold $M$. And one can check $\xi^j$ vary contravariantly under coordinate changes, and therefore define a tangent vector field on $M$ (Einstein's summation convention used): $$X=\xi^j\frac{\partial}{\partial x^j}$$ Then the paper says and I quote:

...whence by the Green's divergence theorem $$\int_M\xi^j_jdM=0.$$

where $dM$ is the volume element on $M$.

There are two things I am not sure of.

  1. I am not sure what $\xi^j_j$ here means. My best guess would be $\xi^j_j=\frac{\partial\xi^j}{\partial x^j}$.
  2. Suppose $\xi^j_j$ does mean as above. What I don't get is why $\int_M\xi^j_jdM=0$ follows from the divergence theorem. According to Proposition 2.46 of Lee's book Introduction to Riemannian Manifolds, the coordinate representation of the divergence of $X=\xi^j\frac{\partial}{\partial x^j}$ should be ($\det g$ is the determinant of the matrix for the metric tensor under local coordinates) $${\rm div} X=\frac{1}{\sqrt{\det g}}\frac{\partial}{\partial x^j}(\xi^j\sqrt{\det g})$$ The divergence theorem then takes the form ($M$ is assumed to be compact without boundary so that the RHS vanish): $$\int_M(\xi^j_j+\frac{\xi^j}{\sqrt{\det g}}\frac{\partial\sqrt{\det g}}{\partial x^j})dM=0$$ There's clearly an extra term.

Any help is appreciated.

  • 6
    $\begingroup$ I haven't read the paper bu I would bet $\xi_j^j$ means $\nabla_j\xi^j$ which is exactly $\mathrm{div}\, X$. $\endgroup$ Jan 16, 2020 at 18:48
  • 1
    $\begingroup$ @ThomasRichard Thank you! I think you are right. I redid the calculation assuming the subscript means covariant derivative and everything works now. $\endgroup$
    – trisct
    Jan 16, 2020 at 19:10


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