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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 Andres Caicedo, Olivier Benoist, j.c., Suvrit, Chris Godsil Nov 25 '13 at 18:30

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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
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Why has the user asking this question been deleted? –  Stefan Kohl Nov 25 '13 at 20:32