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For a more elegant way, suppose given a [[vector bundle]] over M, with ''n''-dimensional [[fiber]] ''V''. Equip this vector bundle with a [[connection form|connection]]. Suppose too we have a [[smooth section]] ''f'' of this bundle.
Then the [[covariant derivative]] of ''f'' with respect to the connection is a smooth [[linear map]] Δ''f'' from the [[tangent bundle]] ''TM'' to ''V'' which preserves the [[base point]]. Assume this linear map is right [[invertible]] (i.e. there exists a linear map ''g'' such that (Δ''f'')''g'' is the [[identity function|identity map]]) for all the fibers at the zeros of ''f''. Then, according to the [[implicit function theorem]], the subspace of zeros of ''f'' is a [[submanifold]].
The ordinary [[Poisson bracket]] is only defined over <math>C^{\infty}(M)</math>, the space of smooth functions over ''M''. However, using the connection, we can extend it to the space of smooth sections of ''f'' if we work with the [[algebra bundle]] with the [[graded algebra]] of ''V''-tensors as fibers. Assume also that under this Poisson bracket,
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