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- Contact manifold and Darboux's theorem for one-form ##\theta## such that ##d\theta## is a 2-form with constant rank 0

Hi, I'm studying the concept of contact manifold -- Contact geometry

A related theorem is Darboux's theorem for one-forms -- Darboux theorem

In the particular case of one-form ##\theta \neq 0## such that ##d\theta## has constant rank 0 then if ##\theta \wedge (d\theta)^0 \neq 0## there exists a local coordinate chart such that ##\theta=dx_1##.

My question is: what does it mean ##d\theta## has rank 0 ? Thanks.

A related theorem is Darboux's theorem for one-forms -- Darboux theorem

In the particular case of one-form ##\theta \neq 0## such that ##d\theta## has constant rank 0 then if ##\theta \wedge (d\theta)^0 \neq 0## there exists a local coordinate chart such that ##\theta=dx_1##.

My question is: what does it mean ##d\theta## has rank 0 ? Thanks.

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