Separate first-order necessary and sufficient conditions are given for local controllability of affine connection control systems. Both conditions are given in terms of a vector-valued quadratic form defined using the symmetric product. The conditions are quite general. For example, they are sharper than the good/bad bracket test of Sussmann when applied to these systems. Indeed, it is conjectured that these are the sharpest possible conditions possible up to first order. This conjecture is partially proven.
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