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quasar987

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## Main Question or Discussion Point

If the tangent bundle is trivial, then the cotangent bundle is trivial. To see this, consider (X_i) a global frame for TM. Then define a global frame (\alpha^i) for T*M by setting [itex]\alpha^i(X_j)=\delta_{ij}[/itex] and extend by linearity.

Does trivial cotangent bundle implies trivial tangent bundle? A similar argument based on global frames does not seem to work in this direction: given a global frame (\alpha^i) for T*M, how do you define a global frame for TM? It does not makse sense to say "Let X_i be the vector field such that [itex]\alpha^i(X_j)=\delta_{ij}[/itex]" because such a vector field might not exist. And if locally, [itex]\alpha^i=\sum_j\alpha^i_jdx^j[/itex], then defining a (global) vector field by setting [itex]X_i:=\sum_j\alpha^i_j\partial_j[/itex] is inconsistent because the coefficients [itex]\alpha^i_j[/itex] do no transform correctly.

Does trivial cotangent bundle implies trivial tangent bundle? A similar argument based on global frames does not seem to work in this direction: given a global frame (\alpha^i) for T*M, how do you define a global frame for TM? It does not makse sense to say "Let X_i be the vector field such that [itex]\alpha^i(X_j)=\delta_{ij}[/itex]" because such a vector field might not exist. And if locally, [itex]\alpha^i=\sum_j\alpha^i_jdx^j[/itex], then defining a (global) vector field by setting [itex]X_i:=\sum_j\alpha^i_j\partial_j[/itex] is inconsistent because the coefficients [itex]\alpha^i_j[/itex] do no transform correctly.