Coframe

In mathematics, a coframe or coframe field on a smooth manifold M {displaystyle M} is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra of M {displaystyle M} , one has a natural map from v k : ⨁ k T ∗ M → ⋀ k T ∗ M {displaystyle v_{k}:igoplus ^{k}T^{*}M o igwedge ^{k}T^{*}M} , given by v k : ( ρ 1 , … , ρ k ) ↦ ρ 1 ∧ … ∧ ρ k {displaystyle v_{k}:( ho _{1},ldots , ho _{k})mapsto ho _{1}wedge ldots wedge ho _{k}} . If M {displaystyle M} is n {displaystyle n} dimensional a coframe is given by a section σ {displaystyle sigma } of ⨁ n T ∗ M {displaystyle igoplus ^{n}T^{*}M} such that v n ∘ σ ≠ 0 {displaystyle v_{n}circ sigma eq 0} . The inverse image under v n {displaystyle v_{n}} of the complement of the zero section of ⋀ n T ∗ M {displaystyle igwedge ^{n}T^{*}M} forms a G L ( n ) {displaystyle GL(n)} principal bundle over M {displaystyle M} , which is called the coframe bundle. In mathematics, a coframe or coframe field on a smooth manifold M {displaystyle M} is a system of one-forms or covectors which form a basis of the cotangent bundle at every point. In the exterior algebra of M {displaystyle M} , one has a natural map from v k : ⨁ k T ∗ M → ⋀ k T ∗ M {displaystyle v_{k}:igoplus ^{k}T^{*}M o igwedge ^{k}T^{*}M} , given by v k : ( ρ 1 , … , ρ k ) ↦ ρ 1 ∧ … ∧ ρ k {displaystyle v_{k}:( ho _{1},ldots , ho _{k})mapsto ho _{1}wedge ldots wedge ho _{k}} . If M {displaystyle M} is n {displaystyle n} dimensional a coframe is given by a section σ {displaystyle sigma } of ⨁ n T ∗ M {displaystyle igoplus ^{n}T^{*}M} such that v n ∘ σ ≠ 0 {displaystyle v_{n}circ sigma eq 0} . The inverse image under v n {displaystyle v_{n}} of the complement of the zero section of ⋀ n T ∗ M {displaystyle igwedge ^{n}T^{*}M} forms a G L ( n ) {displaystyle GL(n)} principal bundle over M {displaystyle M} , which is called the coframe bundle.