In differential geometry, a "'one-form "'on a differentiable manifold is a section of the cotangent bundle.
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Where V _ \ Sigma Y and V _ \ Sigma ^ * Y are the vertical cotangent bundle of Y \ to \ Sigma.
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The 1-form is a section of the cotangent bundle, that gives a local linear approximation to in the cotangent space at each point.
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While the upper bound is an immediate consequence of the above interpretation of in terms of the cotangent bundle, the lower bound is more subtle.
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If is a real differentiable function over, then is a section of the cotangent bundle and as such, we can construct a map from to.
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The exterior derivative of & theta; is a Hamiltonian; thus the cotangent bundle can be understood to be a phase space on which Hamiltonian mechanics plays out.
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Another classical case occurs when M is the cotangent bundle of \ mathbb { R } ^ 3 and G is the Euclidean group generated by rotations and translations.
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Let N be a smooth manifold and let T ^ * N be its cotangent bundle, with projection map \ pi : T ^ * N \ rightarrow N.
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There are more complicated operations on the Floer homology of a cotangent bundle that correspond to the string topology operations on the homology of the loop space of the underlying manifold.
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The cotangent bundle of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the cotangent space . differential one-forms.
