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When I read White's paper (Comment. Math. Helvetici, 1989) named: "A new proof of the compactness theorem for integral currents", I am confused about lemma 2.2, the constant vector field lemma. Can someone tell me what is the definition of a current $T$ is translation invariant along the direction $V$?

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closed as off-topic by Andrés E. Caicedo, Olivier Benoist, j.c., Suvrit, Chris Godsil Nov 25 '13 at 18:30

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – AndrĂ©s E. Caicedo, Olivier Benoist, Chris Godsil
If this question can be reworded to fit the rules in the help center, please edit the question.

I'm not sure why this got closed so quickly, but here is the answer: a current is a linear function on forms. Translation in the direction $V$, say $T_V$ has a natural action on forms by pullback. Thus, to define the translation of a current $C$ in the $V$, we say that $(T_V^*C)(w) := C((T_V)_*w)$. Of course, $C$ being invariant under the translation means that $T_V^*C = C$ as currents (i.e., we have equality when paired with any form). – Otis Chodosh Nov 25 '13 at 18:33
Why has the user asking this question been deleted? – Stefan Kohl Nov 25 '13 at 20:32