Vector space-valued differential 1-form

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 $T_{p}M$, is a linear map from $T_{p}M$ to $V$
• A smoothly varying collection of linear maps from $T_{p}M$ 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.