The cotangent bundle $T^*M$ as the disjoint union $\bigsqcup_{x \in M} T^*_x M$ of cotangent spaces.
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The bundle projection $\pi : T^*M \to M$, $(x, \xi) \mapsto x$.
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Builds a cotangent vector at $x \in M$ from a covector $\xi \in T^*_x M$.
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Projection of a built cotangent vector returns its basepoint.
The natural topology on $T^*M$ making the projection continuous (axiomatized).
$T^*M$ inherits a charted space structure with model $H \times E$ (axiomatized).
$T^*M$ is a smooth manifold with model $I \times \mathrm{id}_E$ (axiomatized).
The model-with-corners for $T^*M$, namely $I \times \mathrm{id}_E$.
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The space of differential $p$-forms on the cotangent bundle $T^*M$.
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The space of vector fields on the cotangent bundle $T^*M$.
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The DifferentialFormSpace structure on $T^*M$, induced from its manifold structure.
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Instance witness for the differential form space structure on the cotangent bundle.
The canonical (Liouville/tautological) $1$-form $\theta$ on $T^*M$, defined locally by $\theta_{(x,\xi)} = \xi \circ d\pi$.
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The canonical symplectic form $-d\theta$ on $T^*M$ is nondegenerate as a pairing on vector fields.
The cotangent bundle $T^*M$ equipped with its canonical symplectic structure $\omega_0 = d\theta$.
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The zero section $M \hookrightarrow T^*M$, $x \mapsto (x, 0)$, as a DFS-morphism.
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The Liouville form pulls back to zero along the zero section: $0^*\theta = 0$.
Assembles all the data above into a CotangentBundleDFS structure for $T^*M$.
Convenience abbreviation for the CotangentBundleDFS structure of $T^*M$.
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The canonical symplectic $2$-form $\omega_0$ on $T^*M$.
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The canonical Liouville $1$-form $\theta$ on $T^*M$.
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The canonical symplectic form on $T^*M$ is exact, $\omega_0 = d\theta$.