Sharp map

In differential geometry, the sharp map is the mapping that converts coordinate 1-forms into corresponding coordinate basis vectors.

Definition

Let $M$ be a manifold and $,Gamma\; (TM)$ denote the space of all sections of the tangent bundle. Fix a nondegenerate (0,2)-tensorfield $g\; in\; Gamma(T^*M^\{otimes\; 2\})$ , i.e a metric tensor or a symplectic form. The definition:$X^flat\; :=\; i\_X\; g\; =\; g(X,.)$yields a linear map sometimes called the flat map:$flat\; :\; Gamma(TM)\; o\; Gamma(T^*M)$which is an isomorphism, since $g$ is non-degenerate. Its inverse:$sharp\; :=\; flat^\{-1\}\; :\; Gamma(T^*M)\; o\; Gamma(TM)$is called the sharp map.