differential geometry cheat sheet ed

Table of Contents

Differential Forms ed

exterior product
\[ \alpha \wedge \beta = (-1)^{\operatorname{deg} \alpha \, \operatorname{deg} \beta} \beta \wedge \alpha \]

exterior derivative
\[ d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^{\operatorname{deg} \alpha} \alpha \wedge d\beta \]
\[ (d\omega)(X_0,\dots,X_k) = \sum_i (-1)^i X_i (\omega(\dots, X_{\neq i}, \dots)) + \sum_{i\lt j} \omega([X_i,X_j], \dots, X_{\neq i,j}, \dots) \]

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

Hodge star
\[ \langle \alpha, \beta \rangle \operatorname{vol} = \alpha \wedge \star \beta \]
\[ \star 1 = \operatorname{vol} \]
\[ \star \star = (-1)^{k(n-k)} (-1)^{\operatorname{sig} g} \]
\[ \langle \operatorname{vol}, \operatorname{vol} \rangle = \star \operatorname{vol} = (-1)^{\operatorname{sig} g} \]
\[ \langle \alpha, \beta \rangle = (-1)^{\operatorname{sig} g} \langle \star \alpha, \star \beta \rangle \]

codifferential
\[ \delta = (-1)^k \star^{-1} d \star \]
\[ (\alpha, \delta \beta) = (d\alpha, \beta) \]
\[ \triangle = (\delta + d)^2 = \delta d + d \delta \]

Covariant Derivative ed

Christoffel symbols
\[ \Gamma^i_{jk} = \frac{1}{2} g^{ia} (g_{aj,k} + g_{ka,j} - g_{jk,a}) = \langle \nabla_{e_i} e_j, e_k \rangle \]

vector field
\[ \nabla_X V = X^i (\partial_i V^j + \Gamma^j_{ik} V^k ) e_j \]

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

Categories: Mathematik, Differentialgeometrie