Sharp map
In differential geometry, the sharp map is the mapping that converts 1-forms into corresponding vectors, given a non-degenerate (0,2)-tensor. Definition. Let formula_1 be a manifold and formula_2 denote the space of all sections of its tangent bundle. Fix a nondegenerate (0,2)-tensor field formula_3 , for example a metric tensor or a symplectic form. The definition yields a linear map sometimes called the flat map which is an isomorphism, since formula_6 is non-degenerate. Its inverse is called the sharp map.