# Sharp map

Jump to navigation
Jump to search

Template:No sources
In differential geometry, the **sharp map** is the mapping that converts coordinate 1-forms into corresponding coordinate basis vectors.

## Definition

Let be a manifold and denote the space of all sections of its tangent bundle. Fix a nondegenerate (0,2)-tensor field , i.e., a metric tensor or a symplectic form. The definition

yields a linear map sometimes called the flat map

which is an isomorphism, since is non-degenerate. Its inverse

is called the sharp map.