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==Geometric theory==
For a more elegant way, suppose given a [[vector bundle]] over <math>\mathcal M</math>, with <math>n</math>-dimensional [[Fiber (mathematics)|fiber]] <math>V</math>. Equip this vector bundle with a [[connection form|connection]]. Suppose too we have a [[Section (fiber bundle)|smooth section]] {{mvar|f}} of this bundle.
Then the [[covariant derivative]] of {{mvar|f}} with respect to the connection is a smooth [[linear map]] <math>\nabla f</math> from the [[tangent bundle]] <math>T\mathcal M</math> to <math>V</math>, which preserves the [[base point]]. Assume this linear map is right [[invertible]] (i.e. there exists a linear map <math>g</math> such that <math>(\Delta f)g</math> is the [[identity function|identity map]]) for all the fibers at the zeros of {{mvar|f}}. Then, according to the [[implicit function theorem]], the subspace of zeros of {{mvar|f}} is a [[submanifold]].
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