Then the covariant derivative of f with respect to the connection is a smooth linear map ∇ ∇ f from the tangent bundle T M 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 Δ f ) g is the 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.