differential geometry cheat sheet ed

Table of Contents

Covariant Derivative ed

vector field
\[ \nabla_X V \]

tensor field \[ A \in \mathfrak T^k_l \]
\[ (\nabla_X A)(Y_1,\dots,Y_k) = \nabla_X(A(Y_1,\dots,Y_k)) - \sum_i A(Y_1,\dots,\nabla_X Y_i,\dots,Y_k) \]
\[ (\nabla_X A)(Y_1,\dots,Y_k,\alpha_1,\dots,\alpha_l) = \nabla_X(A(\dots)) - \sum_i A(\dots,\nabla_X Y_i,\dots) - \sum_i A(\dots,\nabla_X \alpha_i,\dots) \]
\[ \nabla_X ( A \otimes B) = (\nabla_X A) \otimes B + A \otimes (\nabla_X B) \]

curvature
\[ R(X,Y) = \nabla_X \nabla_Y - \nabla_Y \nabla_X - \nabla_{[X,Y]} \]

torsion
\[ T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y] \]

Connection Form ed

basis \[e_i\]

connection form vs. connection
\[ \omega^i_j(X) \, e_i = \nabla_X e_j \]

curvature
\[ R(X,Y) e_k = \Omega^i_k(X,Y) \, e_i \]

vector field
\[ \nabla_X V = X(V^i) e_i + V^j \, \omega^i_j(X) e_i \]

dual vector field
\[ \nabla_X \alpha = X(\alpha_i) e^i - \alpha_i \, \omega^i_j(X) e^j \]

structure equations
\[ d\theta^i = - \omega^i_k \wedge \theta^k \]
\[ d\omega^i_j = - \omega^i_k \wedge \omega^k_j + \Omega^i_j \]

Differential Forms ed

\[ (d\omega)(X_0,\dots,X_k) = \sum_i (-1)^i X_i (\omega(X_{\neq i})) + \sum_{i\lt j} \omega([X_i,X_j], X_{\neq i,j}) \]

\[ \mathcal L_X = i_X \circ d + d \circ i_X \] (Cartan)
\[ \mathcal L_X \circ d + d \circ \mathcal L_X \]

Categories: Mathematik, Differentialgeometrie