Sharp map

Short description: Mathematical technique

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 [math]\displaystyle{ M }[/math] be a manifold and [math]\displaystyle{ \,\Gamma (TM) }[/math] denote the space of all sections of its tangent bundle. Fix a nondegenerate (0,2)-tensor field [math]\displaystyle{ g \in \Gamma(T^*M^{\otimes 2}) }[/math] , for example a metric tensor or a symplectic form. The definition

[math]\displaystyle{ X^\flat := i_X g = g(X,\cdot) }[/math]

yields a linear map sometimes called the flat map

[math]\displaystyle{ \flat : \Gamma(TM) \to \Gamma(T^*M) }[/math]

which is an isomorphism, since [math]\displaystyle{ g }[/math] is non-degenerate. Its inverse

[math]\displaystyle{ \sharp := \flat^{-1} : \Gamma(T^*M) \to \Gamma(TM) }[/math]

is called the sharp map.