Sharp map - Alchetron, The Free Social Encyclopedia

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 Γ ( T M ) denote the space of all sections of its tangent bundle. Fix a nondegenerate (0,2)-tensor field g ∈ Γ ( T ∗ M ⊗ 2 ) , i.e., a metric tensor or a symplectic form. The definition

X ♭ := i X g = g ( X , . )

yields a linear map sometimes called the flat map

♭ : Γ ( T M ) → Γ ( T ∗ M )

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

♯ := ♭ − 1 : Γ ( T ∗ M ) → Γ ( T M )

is called the sharp map.

References

Sharp map Wikipedia

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