Vector space-valued differential 1-form

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Definition

Symbol-free definition

A vector space-valued differential 1-form on a differential manifold, is defined in the following ways:

  • It is a smooth map from the tangent bundle of the manifold, to a given vector space, such that the restriction to the tangent space at any point, is a linear map
  • It is a smoothly varying collection of linear maps from the tangent spaces at different points on the manifold, to a fixed vector space
  • It is a section of the tensor product of the cotangent bundle of the manifold, with a vector space

Definition with symbols

Let M be a differential manifold and V a finite-dimensional real vector space. A V-valued differential 1-form on M is defined as follows:

  • A smooth map from TM to V, whose restriction to each fiber T_pM, is a linear map from T_pM to V
  • A smoothly varying collection of linear maps from T_pM to V for every p \in M
  • A section of the bundle T^*M \otimes V

A particular case of this is a Lie algebra-valued differential 1-form. Here, a Lie group G is acting on the differential manifold M, and the vector space is the Lie algebra \mathfrak{g} of G. Examples are the Maurer-Cartan form and connection form.