In differential geometry, the connection form is a tool used to describe the connection (or covariant derivative) on a principal bundle. It provides a way to differentiate sections of the bundle and understand how they change as they move along the base manifold. Here’s a detailed explanation: ### Principal Bundles A principal bundle is a mathematical structure that consists of: 1. **Base Manifold (M)**: The space over which the bundle is defined. 2. **Total Space (P)**: The larger space that 'fibers' over the base manifold. 3. **Structure Group (G)**: A Lie group that acts on the fibers in a way that preserves the bundle's structure. A principal $G$-bundle $P$ over a base manifold $M$ is denoted $P(M, G)$. ### Connection on a Principal Bundle A connection on a principal bundle provides a way to differentiate along the fibers. It allows for the definition of parallel transport and the covariant derivative. ### Connection Form The connection form is a 1-form on the total space $P$ of the principal bundle that encodes information about the connection. Here’s a more formal definition: #### Definition A connection form $\omega$ on a principal $G$-bundle $P$ is a $\mathfrak{g}$-valued 1-form on $P$ (where $\mathfrak{g}$ is the Lie algebra of $G$) satisfying two properties: 1. **Equivariance**: For any element $g$ in the structure group $G$ and any vector $X$ in the tangent space of $P$, $R_g^* \omega = \text{Ad}(g^{-1}) \omega,$ where $R_g$ is the right action of $g$ on $P$, and $\text{Ad}$ is the adjoint action of $G$ on its Lie algebra $\mathfrak{g}$. 2. **Reproduction of the Lie Algebra**: For any fundamental vector field $\xi^\#$ on $P$ corresponding to an element $\xi$ in $\mathfrak{g}$, $\omega(\xi^\#) = \xi.$ #### Properties - **Equivariance** ensures that the connection form respects the symmetry of the principal bundle. - **Reproduction** ensures that the connection form correctly captures the infinitesimal generators of the group action. ### Usage - **Covariant Derivative**: The connection form allows the definition of a covariant derivative on associated vector bundles, which are bundles constructed from the principal bundle by associating a vector space to each point in the base manifold. - **Parallel Transport**: It enables the definition of parallel transport along paths in the base manifold. - **Curvature**: The curvature of the connection is given by the exterior derivative of the connection form plus a term involving the wedge product of the form with itself, representing the failure of second covariant derivatives to commute. ### Example: The Levi-Civita Connection In the case of the [[Levi-Civita connection]] on the frame bundle of a [[Riemannian manifold]], the connection form can be represented in terms of the Christoffel symbols $\Gamma^i_{jk}$​. Here, the connection form encodes information about how the frame (basis of the tangent space) changes from point to point. ### Summary The connection form is a powerful tool in differential geometry, encapsulating the information about a connection on a principal bundle in a $\mathfrak{g}$-valued 1-form. This allows for a consistent and structured way to differentiate and understand the geometry of the bundle, providing a foundation for concepts like covariant derivatives, parallel transport, and curvature. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Connection form") or contains(subject, "Vector Bundle") sort title, authors, modified, desc ```