The cotangent space at is the dual vector space . Its elements are covectors on the tangent space. In a manifold chart , the basis is dual to . Under a coordinate change to , the coefficients of obey . The disjoint union of these spaces is the cotangent bundle.
A Riemannian metric induces a dual inner product on each cotangent space. On decomposable covectors define and extend bilinearly. Increasing wedges of an orthonormal coframe form an orthonormal basis. This positive-definite inner product is the one used to define the Hodge star operator.
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Cotangent space is a concept from differential geometry and differential topology. It is closely related to the notion of tangent space, which is used to analyze the local properties of smooth manifolds. 1. **Tangent Space**: The tangent space at a point on a manifold consists of the tangent vectors that can be considered as equivalence classes of curves passing through that point, or more abstractly, as derivations acting on smooth functions defined near that point.