Let \(M\) be a smooth manifold. A connection is a map
\begin{equation*} \nabla\colon \mathfrak{X}(M)\times \mathfrak{X}(M) \to \mathfrak{X}(M), \end{equation*}written \(\nabla(X,Y)=\nabla_X Y\), satisfying
- \(C^\infty\) linearity in the firs argument: For \(f\in C^\infty(M)\), \begin{equation*} \nabla_{fX_1+X_2}Y=f\nabla_{X_1}Y+\nabla_{X_2}Y, \end{equation*}
- \(\mathbb{R}\) linearity in the second argument: For \(a\in \mathbb{R}\), \begin{equation*} \nabla_X (aY_1+Y_2)=a\nabla_XY_1+\nabla_X Y_2, \end{equation*}
- product rule: For \(f\in C^\infty(M)\), \begin{equation*} \nabla_X (fY)=f\nabla_XY+X(f)Y. \end{equation*}
Links Link to heading
- connection in local coordinates
- connections in tensor bundles
- covariant derivative along curves
- total covariant derivative